example of citation in research example

APA Citation Style

Citation examples.

• Paper Format
• Style and Grammar Guidelines
• Citation Management Tools
• What's New in the 7th Edition?
• APA Style References Guidelines from the American Psychological Association
• APA Style (OWL - Online Writing Lab, Purdue University)
• Common Reference Examples Handout
• Journal Article
• Magazine Article
• Newspaper Article
• Edited Book Chapter
• Dictionary Entry
• Government Report
• YouTube Video
• Facebook Post
• Webpage on a Website
• Supplemental Reference Examples
• Archival Documents and Collections

Parenthetical citations:  (Grady et al., 2019; Jerrentrup et al., 2018)

Narrative citations:  Grady et al. (2019) and Jerrentrup et al. (2018)

• If a journal article has a DOI, include the DOI in the reference.
• If the journal article does not have a DOI and is from an academic research database, end the reference after the page range (for an explanation of why, see the  database information  page). The reference in this case is the same as for a print journal article.
• Do not include database information in the reference unless the journal article comes from a database that publishes original, proprietary content, such as UpToDate (see an example on the  database information  page).
• If the journal article does not have a DOI but does have a URL that will resolve for readers (e.g., it is from an online journal that is not part of a database), include the URL of the article at the end of the reference.
• If the journal article has an article number instead of a page range, include the article number instead of the page range (as shown in the Jerrentrup et al. example).

Parenthetical citations:  (Rabinowitz, 2019; Sapolsky, 2017)

Narrative citations:  Rabinowitz (2019) and Sapolsky (2017)

• If the book includes a DOI, include the DOI in the reference after the publisher name.
• Do not include the publisher location.
• If the book does not have a DOI and comes from an academic research database, end the book reference after the publisher name. Do not include  database information  in the reference. The reference in this case is the same as for a print book.

Parenthetical citations:  (Schaefer & Shapiro, 2019; Schulman, 2019)

Narrative citations:  Schaefer and Shapiro (2019) and Schulman (2019)

• If a magazine article has a DOI, include the DOI in the reference.
• If the magazine article does not have a DOI and is from an academic research database, end the reference after the page range. Do not include  database information  in the reference. The reference in this case is the same as for a print magazine article.
• If the magazine article does not have a DOI but does have a URL that will resolve for readers (e.g., it is from an online magazine that is not part of a database), include the URL of the article at the end of the reference.
• If the magazine article does not have volume, issue, and/or page numbers (e.g., because it is from an online magazine), omit the missing elements from the reference (as in the Schulman example).

Parenthetical citation:  (Carey, 2019)

Narrative citation:  Carey (2019)

• If the newspaper article is from an academic research database, end the reference after the page range. Do not include  database information  in the reference. The reference in this case is the same as for a print newspaper article.
• If the newspaper article has a URL that will resolve for readers (e.g., it is from an online newspaper), include the URL of the article at the end of the reference.
• If the newspaper article does not have volume, issue, and/or page numbers (e.g., because it is from an online newspaper), omit the missing elements from the reference, as shown in the example.
• If the article is from a news website (e.g., CNN, HuffPost)—one that does not have an associated daily or weekly newspaper—use the format for a  webpage on a website  instead.

Parenthetical citation:  (Aron et al., 2019)

Narrative citation:  Aron et al. (2019)

• If the edited book chapter includes a DOI, include the chapter DOI in the reference after the publisher name.
• If the edited book chapter does not have a DOI and comes from an academic research database, end the edited book chapter reference after the publisher name. Do not include  database information  in the reference. The reference in this case is the same as for a print edited book chapter.
• Do not create references for chapters of authored books. Instead, write a reference for the whole book and cite the chapter in the text if desired (e.g., Kumar, 2017, Chapter 2).

Parenthetical citation:  (Merriam-Webster, n.d.)

Narrative citation:  Merriam-Webster (n.d.)

• Because entries in  Merriam-Webster’s Dictionary  are updated over time and are not archived, include a  retrieval date  in the reference.
• Merriam-Webster is both the author and the publisher, so the name appears in the author element only to avoid repetition.
• To quote a dictionary definition, view the pages on quotations and  how to quote works without page numbers  for guidance. Additionally, here is an example:  Culture  refers to the “customary beliefs, social forms, and material traits of a racial, religious, or social group” (Merriam-Webster, n.d., Definition 1a).

Parenthetical citation:  (National Cancer Institute, 2019)

Narrative citation:  National Cancer Institute (2019)

The specific agency responsible for the report appears as the author. The names of parent agencies not present in the  group author name  appear in the source element as the publisher. This creates concise in-text citations and complete reference list entries.

Parenthetical citation:  (Harvard University, 2019)

Narrative citation:  Harvard University (2019)

• Use the name of the account that uploaded the video as the author.
• If the account did not actually create the work, explain this in the text if it is important for readers to know. However, if that would mean citing a source that appears unauthoritative, you might also look for the author’s YouTube channel, official website, or other social media to see whether the same video is available elsewhere.

Parenthetical citations:  (APA Databases, 2019; Gates, 2019)

Narrative citations:  APA Databases (2019) and Gates (2019)

• Present the name of the individual or group author the same as you would for any other reference. Then provide the Twitter handle (beginning with the @ sign) in square brackets, followed by a period.
• Provide the first 20 words of the tweet as the title. Count a URL, a hashtag, or an emoji as one word each, and include them in the reference if they fall within the first 20 words.
• If the tweet includes an image, a video, a poll, or a thumbnail image with a link, indicate that in brackets after the title: [Image attached], [Video attached], [Thumbnail with link attached].
• The same format used for Twitter is also used for Instagram.  

Parenthetical citation:  (News From Science, 2019)

Narrative citation:  News From Science (2019)

• Provide the first 20 words of the Facebook post as the title. Count a URL or other link, a hashtag, or an emoji as one word each, and include them in the reference if they fall within the first 20 words. 
• If a status update includes images, videos, thumbnail links to outside sources, or content from another Facebook post (such as when sharing a link), indicate that in square brackets.

Parenthetical citations:  (Fagan, 2019; National Institute of Mental Health, 2018; Woodyatt, 2019; World Health Organization, 2018)

Narrative citations:  Fagan (2019), National Institute of Mental Health (2018), Woodyatt (2019), and World Health Organization (2018)

• Provide as specific a  date  as is available on the webpage. This might be a year only; a year and month; or a year, month, and day.
• Italicize the title of a webpage.
• When the author of the webpage and the publisher of the website are the same, omit the publisher name to avoid repetition (as in the World Health Organization example).
• When contents of a page are meant to be updated over time but are not archived, include a  retrieval date  in the reference (as in the Fagan example).
• Use the webpage on a website format for articles from news websites such as CNN and HuffPost (these sites do not have associated daily or weekly newspapers). Use the  newspaper article category  for articles from newspaper websites such as  The New York Times  or  The Washington Post .
• Create a reference to an open educational resources (OER) page only when the materials are available for download directly (i.e., the materials are on the page and/or can be downloaded as PDFs or other files). If you are directed to another website, create a reference to the specific webpage on that website where the materials can be retrieved. Use this format for material in any OER repository, such as OER Commons, OASIS, or MERLOT.
• Do not create a reference or in-text citation for a whole website. To mention a website in general, and not any particular information on that site, provide the name of the website in the text and include the URL in parentheses. For example, you might mention that you used a website to create a survey.

The following supplemental example references are mention in the  Publication Manual:

• retracted journal or magazine article
• edition of the Diagnostic and Statistical Manual of Mental Disorders (DSM)
• edition of the International Statistical Classification of Diseases and Related Health Problems (ICD)
• religious work
• annotated religious work

Archival document and collections are not presented in the  APA Publication Manual, Seventh Edition . This content is available only on the APA Style website .  This guidance has been expanded from the 6th edition.

Archival sources include letters, unpublished manuscripts, limited-circulation brochures and pamphlets, in-house institutional and corporate documents, clippings, and other documents, as well as such nontextual materials as photographs and apparatus, that are in the personal possession of an author, form part of an institutional collection, or are stored in an archive such as the Archives of the History of American Psychology at the University of Akron or the APA Archives. For any documents like these that are available on the open web or via a database (subscription or nonsubscription), follow the reference templates shown in Chapter 10 of the Publication Manual.

The general format for the reference for an archival work includes the author, date, title, and source. The reference examples shown on this page may be modified for collections requiring more or less specific information to locate materials, for different types of collections, or for additional descriptive information (e.g., a translation of a letter). Authors may choose to list correspondence from their own personal collections, but correspondence from other private collections should be listed only with the permission of the collector.

Keep in mind the following principles when creating references to archival documents and collections:

• As with any reference, the purpose is to direct readers to the source, despite the fact that only a single copy of the document may be available and readers may have some difficulty actually seeing a copy.
• Include as much information as is needed to help locate the item with reasonable ease within the repository. For items from collections with detailed finding aids, the name of the collection may be sufficient; for items from collections without finding aids, more information (e.g., call number, box number, file name or number) may be necessary to help locate the item.
• If several letters are cited from the same collection, list the collection as a reference and provide specific identifying information (author, recipient, and date) for each letter in the in-text citations (see Example 3).
• Use square brackets to indicate information that does not appear on the document.
• Use “ca.” (circa) to indicate an estimated date (see Example 5).
• Use italics for titles of archival documents and collections; if the work does not have a title, provide a description in square brackets without italics.
• Separate elements of the source (e.g., the name of a repository, library, university or archive, and the location of the university or archive) with commas. End the source with a period.
• If a publication of limited circulation is available in libraries, the reference may be formatted as usual for published material, without the archival source.
• Note that private letters (vs. those in an archive or repository) are considered personal communications and cited in the text only.

1. Letter from a repository

Frank, L. K. (1935, February 4). [Letter to Robert M. Ogden]. Rockefeller Archive Center (GEB Series 1.3, Box 371, Folder 3877), Tarrytown, NY, United States.

• Parenthetical citation: (Frank, 1935)
• Narrative citation: Frank (1935)
• Because the letter does not have a title, provide a description in square brackets.

2. Letter from a private collection

Zacharius, G. P. (1953, August 15). [Letter to William Rickel (W. Rickel, Trans.)]. Copy in possession of Hendrika Vande Kemp.

• Parenthetical citation: (Zacharius, 1953)
• Narrative citation: Zacharius (1953)
• In this example, Hendrika Vande Kemp is either the author of the paper or the author of the paper has received permission from Hendrika Vande Kemp to cite a letter in Vande Kemp’s private collection in this way. Otherwise, cite a private letter as a  personal communication .

3. Collection of letters from an archive

Allport, G. W. (1930–1967). Correspondence. Gordon W. Allport Papers (HUG 4118.10), Harvard University Archives, Cambridge, MA, United States.

• Parenthetical citation: (Allport, 1930–1967)
• Narrative citation: Allport (1930–1967)

To cite specific letters in the text, provide the author and range of years as shown in the reference list entry, plus details about who wrote the specific letter to whom and when the specific letter was written.

• Parenthetical citation: (Allport, 1930–1967, G. Boring to Allport, December 26, 1937)
• Narrative citation: Allport (1930–1967, Allport to G. Boring, March 1, 1939)
• Use the parenthetical citation format to cite a letter that E. G. Boring wrote to Allport because Allport is the author in the reference. Use either the parenthetical or narrative citation format to cite letters that Allport wrote.

4. Unpublished papers, lectures from an archive or personal collection

Berliner, A. (1959). Notes for a lecture on reminiscences of Wundt and Leipzig. Anna Berliner Memoirs (Box M50), Archives of the History of American Psychology, University of Akron, Akron, OH, United States.

• Parenthetical citation: (Berliner, 1959)
• Narrative citation: Berliner (1959)

5. Archival/historical source for which the author and/or date is known or is reasonably certain but not stated on the document

Allport, A. (presumed). (ca. 1937). Marion Taylor today—by the biographer [Unpublished manuscript]. Marion Taylor Papers, Schlesinger Library, Radcliffe College, Cambridge, MA, United States.

• Parenthetical citation: (Allport, ca. 1937)
• Narrative citation: Allport (ca. 1937)
• Because the author is reasonably certain but not stated on the document, place the word “presumed” in parentheses after the name, followed by a period.
• Because the date is reasonably certain but not stated on the document, the abbreviation “ca.” (which stands for “circa”) appears before the year in parentheses.

6. Archival source with group author

Subcommittee on Mental Hygiene Personnel in School Programs. (1949, November 5–6). Meeting of Subcommittee on Mental Hygiene Personnel in School Programs. David Shakow Papers (M1360), Archives of the History of American Psychology, University of Akron, Akron, OH, United States.

• Parenthetical citation: (Subcommittee on Mental Hygiene Personnel in School Programs, 1949)
• Narrative citation: Subcommittee on Mental Hygiene Personnel in School Programs (1949)

7. Interview recorded and available in an archive

Smith, M. B. (1989, August 12). Interview by C. A. Kiesler [Tape recording]. President’s Oral History Project, American Psychological Association, APA Archives, Washington, DC, United States.

• Parenthetical citation: (Smith, 1989)
• Narrative citation: Smith (1989)
• For interviews and oral histories recorded in an archive, list the interviewee as the author. Include the interviewer’s name in the description.

8. Transcription of a recorded interview, no recording available

Sparkman, C. F. (1973). An oral history with Dr. Colley F. Sparkman/Interviewer: Orley B. Caudill. Mississippi Oral History Program (Vol. 289), University of Southern Mississippi, Hattiesburg, MS, United States.

• Parenthetical citation: (Sparkman, 1973)
• Narrative citation: Sparkman (1973)

9. Newspaper article clipping, historical, in personal collection

Psychoanalysis institute to open. (1948, September 18). [Clipping from an unidentified Dayton, OH, United States, newspaper]. Copy in possession of author.

• Parenthetical citation: (“Psychoanalysis Institute to Open,” 1948)
• Narrative citation: “Psychoanalysis Institute to Open” (1948)
• Use this format only if you are the person who is in possession of the newspaper clipping.

10. Historical publication of limited circulation

Sci-Art Publishers. (1935). Sci-Art publications [Brochure]. Roback Papers (HUGFP 104.50, Box 2, Folder “Miscellaneous Psychological Materials”), Harvard University Archives, Cambridge, MA, United States.

• Parenthetical citation: (Sci-Art Publishers, 1935)
• Narrative citation: Sci-Art Publishers (1935)

11. Archived photographs, no author and no title

[Photographs of Robert M. Yerkes]. (ca. 1917–1954). Robert Mearns Yerkes Papers (Box 137, Folder 2292), Manuscripts and Archives, Yale University Library, New Haven, CT, United States.

• Parenthetical citation: ([Photographs of Robert M. Yerkes], ca. 1917–1954)
• Narrative citation: [Photographs of Robert M. Yerkes] (ca. 1917–1954)
• Because the archived photographs do not have a title, provide a bracketed description instead.
• Because the archived photographs do not have an author, move the bracketed description to the author position of the reference.

12. Microfilm

U.S. Census Bureau. (1880). 1880 U.S. census: Defective, dependent, and delinquent classes schedule: Virginia [Microfilm]. NARA Microfilm Publication T1132 (Rolls 33–34), National Archives and Records Administration, Washington, DC, United States.

• Parenthetical citation: (U.S. Census Bureau, 1880)
• Narrative citation: U.S. Census Bureau (1880)

Read the full APA guidelines on citing ChatGPT 

OpenAI. (2023).  ChatGPT  (Mar 14 version) [Large language model].  https://chat.openai.com/chat

• Parenthetical citation:  (OpenAI, 2023)
• Narrative citation:  OpenAI (2023)

Author:  The author of the model is OpenAI.

Date:  The date is the year of the version you used. Following the template in Section 10.10, you need to include only the year, not the exact date. The version number provides the specific date information a reader might need.

Title:  The name of the model is “ChatGPT,” so that serves as the title and is italicized in your reference, as shown in the template. Although OpenAI labels unique iterations (i.e., ChatGPT-3, ChatGPT-4), they are using “ChatGPT” as the general name of the model, with updates identified with version numbers.

The version number is included after the title in parentheses. The format for the version number in ChatGPT references includes the date because that is how OpenAI is labeling the versions. Different large language models or software might use different version numbering; use the version number in the format the author or publisher provides, which may be a numbering system (e.g., Version 2.0) or other methods.

Bracketed text  is used in references for additional descriptions when they are needed to help a reader understand what’s being cited. References for a number of common sources, such as journal articles and books, do not include bracketed descriptions, but things outside of the typical peer-reviewed system often do. In the case of a reference for ChatGPT, provide the descriptor “Large language model” in square brackets. OpenAI describes ChatGPT-4 as a “large multimodal model,” so that description may be provided instead if you are using ChatGPT-4. Later versions and software or models from other companies may need different descriptions, based on how the publishers describe the model. The goal of the bracketed text is to briefly describe the kind of model to your reader.

Source:  When the publisher name and the author name are the same, do not repeat the publisher name in the source element of the reference, and move directly to the URL. This is the case for ChatGPT. The URL for ChatGPT is  https://chat.openai.com/chat . For other models or products for which you may create a reference, use the URL that links as directly as possible to the source (i.e., the page where you can access the model, not the publisher’s homepage).

What to include and what to exclude

Works included in a reference list.

The reference list provides a reliable way for readers to identify and locate the works cited in a paper. APA Style papers generally include reference lists, not  bibliographies.

In general, each work cited in the text must appear in the reference list, and each work in the reference list must be cited in the text. Check your work carefully before submitting your manuscript or course assignment to ensure no works cited in the text are missing from the reference list and vice versa, with only the following exceptions.

Works Excluded From a Reference List

There are a few kinds of works that are not included in a reference list. Usually a work is not included because readers cannot recover it or because the mention is so broad that readers do not need a reference list entry to understand the use.

Information on works included in a reference list is covered in Sections 2.12 and 8.4 of the  APA Publication Manual, Seventh Edition

*This guidance has been expanded from the 6th edition.*

• Personal communications  such as emails, phone calls, or text messages are cited in the text only, not in the reference list, because readers cannot retrieve personal communications.
• General mentions of whole websites, whole periodicals, and common software and apps in the text do not require in-text citations or reference list entries because the use is broad and the source is familiar.
• The source of an epigraph does not usually appear in the reference list unless the work is a scholarly book or journal. For example, if you open the paper with an inspirational quotation by a famous person, the source of the quotation does not appear in the reference list because the quotation is meant to set the stage for the work, not substantiate a key point.   
• Quotations from research participants in a study you conducted can be presented and discussed in the text but do not need citations or reference list entries. Citations and reference list entries are not necessary because the quotations are part of your original research. They could also compromise participants’ confidentiality, which is an ethical violation.
• References included in a meta-analysis, which are marked with an asterisk in the reference list, may be cited in the text (or not) at the author’s discretion. This exception is relevant only to authors who are conducting a meta-analysis.

DOIs and URLs

The DOI or URL is the final component of a reference list entry. Because so much scholarship is available and/or retrieved online, most reference list entries end with either a DOI or a URL.

• A DOI is a unique alphanumeric string that identifies content and provides a persistent link to its location on the internet. DOIs can be found in database records and the reference lists of published works.
• A URL specifies the location of digital information on the internet and can be found in the address bar of your internet browser. URLs in references should link directly to the cited work when possible.

Follow these guidelines for including DOIs and URLs in references:

• Include a DOI for all works that have a DOI, regardless of whether you used the online version or the print version.
• If a print work does not have a DOI, do not include any DOI or URL in the reference.
• If an online work has both a DOI and a URL, include only the DOI.
• For works without DOIs from websites (not including academic research databases), provide a URL in the reference (as long as the URL will work for readers).
• For works without DOIs from most  academic research databases , do not include a URL or database information in the reference because these works are widely available. The reference should be the same as the reference for a print version of the work.
• For works from databases that publish original, proprietary material available only in that database (such as the UpToDate database) or for works of limited circulation in databases (such as monographs in the ERIC database), include the name of the database or archive and the URL of the work. If the URL requires a login or is session-specific (meaning it will not resolve for readers), provide the URL of the database or archive home page or login page instead of the URL for the work. See the page on including  database information in references  for more information. 
• If the URL is no longer working or no longer provides readers access to the content you intend to cite, follow the guidance for works with  no source .
• Other alphanumeric identifiers such as the International Standard Book Number (ISBN) and the International Standard Serial Number (ISSN) are not included in APA Style references.

Follow these guidelines to format DOIs and URLs:

• Present both DOIs and URLs as hyperlinks (i.e., beginning with “http:” or “https:”).
• Because a hyperlink leads readers directly to the content, it is not necessary to include the words “Retrieved from” or “Accessed from” before a DOI or URL.
• It is acceptable to use either the default display settings for hyperlinks in your word-processing program (e.g., usually blue font, underlined) or plain text that is not underlined.
• Leave links live if the work is to be published or read online.
• Follow the current recommendations of the International DOI Foundation to format DOIs in the reference list, which as of this publication is as follows:

https://doi.org/ xxxxx

• The string “https://doi.org/” is a way of presenting a DOI as a link, and “xxxxx” refers to the DOI number.
• The preferred format of the DOI has changed over time. Although older works use previous formats (e.g., “http:/dx.doi.org/” or “doi:” or “DOI:” before the DOI number), in your reference list, standardize DOIs into the current preferred format for all entries. For example, use  https://doi.org/10.1037/a0040251  in your reference even though that article, published in 2016, presented the number in an older format.
• Copy and paste the DOI or URL from your web browser directly into your reference list to avoid transcription errors. Do not change the capitalization or punctuation of the DOI or URL. Do not add line breaks manually to the hyperlink; it is acceptable if your word-processing program automatically adds a break or moves the hyperlink to its own line.
• Do not add a period after the DOI or URL because this may interfere with link functionality.

When a DOI or URL is long or complex, you may use shortDOIs or shortened URLs if desired.

• Use the  shortDOI service  provided by the International DOI Foundation to create shortDOIs. A work can have only one DOI and only one shortDOI; the shortDOI service will either produce a new shortDOI for a work that has never had one or retrieve an existing shortDOI.
• Some websites provide their own branded shortened URLs, and independent URL shortening services are available as well. Any shortened URL is acceptable in a reference as long as you check the link to ensure that it takes you to the correct location.
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Citing Sources: APA Citation Examples

• Citing Sources Overview
• Citing in the Sciences & Engineering
• APA Citation Examples
• Chicago Citation Examples
• Biologists: Council of Science Editors (CSE) Examples
• MLA Citation Examples
• Bluebook - Legal Citation
• Citing Orally in Speeches
• Citation Managers

APA Citations

• Periodicals

Basic Format for a Book:

Reference List: Authors' Last name, First Initial. (Year). Book title: Subtitle. (Edition) [if other than the 1st]. Publisher.

In-text: (Author, Year)

   ~ Book with One Author:

Reference List:  Brader, T. (2006). Campaigning for hearts and minds: How emotional appeals in political ads work . University of Chicago Press. 

In-text: (Brader, 2006)

   ~ ​Book with Two  Authors:

Reference List:   Miller, T. E., & Schuh, J. H. (2005). Promoting reasonable expectations: Aligning student and institutional views of the college experience. Jossey-Bass.

In-text:  (Miller & Schuh, 2005) *for more than two authors (3 or more), list only the first author’s name followed by “et al.” in every citation, even the first, unless doing so would create ambiguity between different sources. Example: (Kernis et al., 1993)

Basic format for an eBook:

Reference list:  author's last name, first initial. (year).  book title [format of book]. publisher. url , in-text:  (author, year),   ~ example:, reference list:  brock, j., & arciuli, j. (2014).  communication in autism [ebook edition] .  john benjamins publishing company. https://doi.org/10.1007/978-1-4757-4806-2, in-text:  (brock & arciuli, 2014).

Basic Format for a Print Journal Article: 

Last name, First Initial. (Year, Month Day). Article title.  Magazine/Journal/Newspaper Title ,  Volume number (Issue number), Page numbers of the entire article.

   ~ Example:

Newman, J. L., Fuqua, D. R., Gray, E. A., & Simpson, D. B. (2006). Gender differences in the relationship of anger and depression in a clinical sample.  Journal of Counseling & Development ,  84 , 157-161.

Basic Format for an Online Journal Article:

Author’s Last Name, First Initial. (Year). Article title. Magazine/Journal/Newspaper Title, Volume number (Issue number), Page numbers. doi or URL of publication home page

   ~ Online Journal Article with DOI Assigned:

Basic Format: 

Author, A. A., & Author, B. B. (Year). Title of article.  Title of Journal, volume number (issue number),   page range. https://doi.org/10.0000/0000

Denhart, H. (2008). Deconstructing barriers: Perceptions of students labeled with learning disabilities in higher education.  Journal of Learning Disabilities ,  41 (6), 483-497. https://doi.org/ 10.1177/0022219408321151

    ~ Online Journal Article with no DOI Assigned:

Basic Format:

Author, A. A., & Author, B. B. (Year). Title of article.  Title of Journal, volume number (issue number).  http://www.journalhomepage.com/full/url/

Example: 

von Busch, O., & Palmas, K. (2016). Designing consent: Can design thinking manufacture democratic capitalism?  Organizational Aesthetics, 5 (2), 10-24. http://digitalcommons.wpi.edu/oa/. 

Basic Format for citing an image in the Reference List:

Last name, first initial. (year image was created).  title of work  [type of work]. url , note: if you can only find the screen name of an author, use that as the author's name. maintain the formatting of the screen name. for example, if a screen name is in all lower case, keep the name in lower case in your citations. if there is no title, create your own title that describes the content of the image., example of a reference list citation for an image: , sipler , d. (2005).  nap time [photograph]. flickr.  https://www.flickr.com/photos/photofarmer/284159867/in/set-72157594353612286, formatting figures in your paper:, each image in your paper should have a figure number, a title, and a caption. the caption should describe the image, provide a citation for the image, and provide copyright information. for example:, two cats resting,            , note. this photo shows two orange cats resting in the "loaf" position. from  nap time [photograph], by d. sipler, 2005, flickr ( https://www.flickr.com/photos/photofarmer/284159867/in/set-72157594353612286 ).  cc by 2.0 ., if you have taken the photo or created the image, you do not need to cite it or provide copyright information for it. you will still need to label the picture with a figure number and title, and you will need to provide a caption with information on what the image shows. , for more information on formatting tables and figures in your apa style paper, see:, apa style guide: tables and figures, apa style guide: clip art or stock image references, navigating copyright for reproduced images, if you did not create the image, you need to provide a copyright statement for that image. the apa style blog takes you through the four steps of navigating copyright for reproduced images:, understand the copyright status of the image., determine whether permission is needed to reproduce the image., secure permission to reproduce the image, if permission is needed.  , write the apa style copyright statement and reference list entry for the image.  , for more information on copyright and finding safe to reuse images, see the library's copyright guide . , basic format for a print article: ,    ~ magazine article:,  white, c. (2006, april). the spirit of disobedience: an invitation to resistance. harper's magazine, 312 (1871), 31-40. ,    ~ newspaper article: , zernike, k. (2015, october 25). white house moves to limit school testing.  new york times , p. a1. , note: for newspaper articles,  p. or pp. precedes page numbers for a newspaper reference in apa style. single pages take p., e.g., p. b2; multiple pages take pp., e.g., pp. b2, b4 or pp. c1, c3-c4. ,    ~ newspaper article found on a newspaper's website:, author, a. a. (year, month day). title of article. title of newspaper.  http://www.someaddress.com/full/url/ , zernike, k. (2016, february 29). testing for joy and grit schools nationwide push to measure students’ emotional skills.  the new york times . http://www.nytimes.com/2016/03/01/us/testing-for-joy-and-grit-schools-nationwide-push-to-measure-students-emotional-skills.html_r=.

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Home / Guides / Citation Guides / How to Cite Sources

How to Cite Sources

Here is a complete list for how to cite sources. Most of these guides present citation guidance and examples in MLA, APA, and Chicago.

If you’re looking for general information on MLA or APA citations , the EasyBib Writing Center was designed for you! It has articles on what’s needed in an MLA in-text citation , how to format an APA paper, what an MLA annotated bibliography is, making an MLA works cited page, and much more!

MLA Format Citation Examples

The Modern Language Association created the MLA Style, currently in its 9th edition, to provide researchers with guidelines for writing and documenting scholarly borrowings.  Most often used in the humanities, MLA style (or MLA format ) has been adopted and used by numerous other disciplines, in multiple parts of the world.

MLA provides standard rules to follow so that most research papers are formatted in a similar manner. This makes it easier for readers to comprehend the information. The MLA in-text citation guidelines, MLA works cited standards, and MLA annotated bibliography instructions provide scholars with the information they need to properly cite sources in their research papers, articles, and assignments.

• Book Chapter
• Conference Paper
• Documentary
• Encyclopedia
• Google Images
• Kindle Book
• Memorial Inscription
• Museum Exhibit
• Painting or Artwork
• PowerPoint Presentation
• Sheet Music
• Thesis or Dissertation
• YouTube Video

APA Format Citation Examples

The American Psychological Association created the APA citation style in 1929 as a way to help psychologists, anthropologists, and even business managers establish one common way to cite sources and present content.

APA is used when citing sources for academic articles such as journals, and is intended to help readers better comprehend content, and to avoid language bias wherever possible. The APA style (or APA format ) is now in its 7th edition, and provides citation style guides for virtually any type of resource.

Chicago Style Citation Examples

The Chicago/Turabian style of citing sources is generally used when citing sources for humanities papers, and is best known for its requirement that writers place bibliographic citations at the bottom of a page (in Chicago-format footnotes ) or at the end of a paper (endnotes).

The Turabian and Chicago citation styles are almost identical, but the Turabian style is geared towards student published papers such as theses and dissertations, while the Chicago style provides guidelines for all types of publications. This is why you’ll commonly see Chicago style and Turabian style presented together. The Chicago Manual of Style is currently in its 17th edition, and Turabian’s A Manual for Writers of Research Papers, Theses, and Dissertations is in its 8th edition.

Citing Specific Sources or Events

• Declaration of Independence
• Gettysburg Address
• Martin Luther King Jr. Speech
• President Obama’s Farewell Address
• President Trump’s Inauguration Speech
• White House Press Briefing

Additional FAQs

• Citing Archived Contributors
• Citing a Blog
• Citing a Book Chapter
• Citing a Source in a Foreign Language
• Citing an Image
• Citing a Song
• Citing Special Contributors
• Citing a Translated Article
• Citing a Tweet

6 Interesting Citation Facts

The world of citations may seem cut and dry, but there’s more to them than just specific capitalization rules, MLA in-text citations , and other formatting specifications. Citations have been helping researches document their sources for hundreds of years, and are a great way to learn more about a particular subject area.

Ever wonder what sets all the different styles apart, or how they came to be in the first place? Read on for some interesting facts about citations!

1. There are Over 7,000 Different Citation Styles

You may be familiar with MLA and APA citation styles, but there are actually thousands of citation styles used for all different academic disciplines all across the world. Deciding which one to use can be difficult, so be sure to ask you instructor which one you should be using for your next paper.

2. Some Citation Styles are Named After People

While a majority of citation styles are named for the specific organizations that publish them (i.e. APA is published by the American Psychological Association, and MLA format is named for the Modern Language Association), some are actually named after individuals. The most well-known example of this is perhaps Turabian style, named for Kate L. Turabian, an American educator and writer. She developed this style as a condensed version of the Chicago Manual of Style in order to present a more concise set of rules to students.

3. There are Some Really Specific and Uniquely Named Citation Styles

How specific can citation styles get? The answer is very. For example, the “Flavour and Fragrance Journal” style is based on a bimonthly, peer-reviewed scientific journal published since 1985 by John Wiley & Sons. It publishes original research articles, reviews and special reports on all aspects of flavor and fragrance. Another example is “Nordic Pulp and Paper Research,” a style used by an international scientific magazine covering science and technology for the areas of wood or bio-mass constituents.

4. More citations were created on  EasyBib.com  in the first quarter of 2018 than there are people in California.

The US Census Bureau estimates that approximately 39.5 million people live in the state of California. Meanwhile, about 43 million citations were made on EasyBib from January to March of 2018. That’s a lot of citations.

5. “Citations” is a Word With a Long History

The word “citations” can be traced back literally thousands of years to the Latin word “citare” meaning “to summon, urge, call; put in sudden motion, call forward; rouse, excite.” The word then took on its more modern meaning and relevance to writing papers in the 1600s, where it became known as the “act of citing or quoting a passage from a book, etc.”

6. Citation Styles are Always Changing

The concept of citations always stays the same. It is a means of preventing plagiarism and demonstrating where you relied on outside sources. The specific style rules, however, can and do change regularly. For example, in 2018 alone, 46 new citation styles were introduced , and 106 updates were made to exiting styles. At EasyBib, we are always on the lookout for ways to improve our styles and opportunities to add new ones to our list.

Why Citations Matter

Here are the ways accurate citations can help your students achieve academic success, and how you can answer the dreaded question, “why should I cite my sources?”

They Give Credit to the Right People

Citing their sources makes sure that the reader can differentiate the student’s original thoughts from those of other researchers. Not only does this make sure that the sources they use receive proper credit for their work, it ensures that the student receives deserved recognition for their unique contributions to the topic. Whether the student is citing in MLA format , APA format , or any other style, citations serve as a natural way to place a student’s work in the broader context of the subject area, and serve as an easy way to gauge their commitment to the project.

They Provide Hard Evidence of Ideas

Having many citations from a wide variety of sources related to their idea means that the student is working on a well-researched and respected subject. Citing sources that back up their claim creates room for fact-checking and further research . And, if they can cite a few sources that have the converse opinion or idea, and then demonstrate to the reader why they believe that that viewpoint is wrong by again citing credible sources, the student is well on their way to winning over the reader and cementing their point of view.

They Promote Originality and Prevent Plagiarism

The point of research projects is not to regurgitate information that can already be found elsewhere. We have Google for that! What the student’s project should aim to do is promote an original idea or a spin on an existing idea, and use reliable sources to promote that idea. Copying or directly referencing a source without proper citation can lead to not only a poor grade, but accusations of academic dishonesty. By citing their sources regularly and accurately, students can easily avoid the trap of plagiarism , and promote further research on their topic.

They Create Better Researchers

By researching sources to back up and promote their ideas, students are becoming better researchers without even knowing it! Each time a new source is read or researched, the student is becoming more engaged with the project and is developing a deeper understanding of the subject area. Proper citations demonstrate a breadth of the student’s reading and dedication to the project itself. By creating citations, students are compelled to make connections between their sources and discern research patterns. Each time they complete this process, they are helping themselves become better researchers and writers overall.

When is the Right Time to Start Making Citations?

Make in-text/parenthetical citations as you need them.

As you are writing your paper, be sure to include references within the text that correspond with references in a works cited or bibliography. These are usually called in-text citations or parenthetical citations in MLA and APA formats. The most effective time to complete these is directly after you have made your reference to another source. For instance, after writing the line from Charles Dickens’ A Tale of Two Cities : “It was the best of times, it was the worst of times…,” you would include a citation like this (depending on your chosen citation style):

(Dickens 11).

This signals to the reader that you have referenced an outside source. What’s great about this system is that the in-text citations serve as a natural list for all of the citations you have made in your paper, which will make completing the works cited page a whole lot easier. After you are done writing, all that will be left for you to do is scan your paper for these references, and then build a works cited page that includes a citation for each one.

Need help creating an MLA works cited page ? Try the MLA format generator on EasyBib.com! We also have a guide on how to format an APA reference page .

2. Understand the General Formatting Rules of Your Citation Style Before You Start Writing

While reading up on paper formatting may not sound exciting, being aware of how your paper should look early on in the paper writing process is super important. Citation styles can dictate more than just the appearance of the citations themselves, but rather can impact the layout of your paper as a whole, with specific guidelines concerning margin width, title treatment, and even font size and spacing. Knowing how to organize your paper before you start writing will ensure that you do not receive a low grade for something as trivial as forgetting a hanging indent.

Don’t know where to start? Here’s a formatting guide on APA format .

3. Double-check All of Your Outside Sources for Relevance and Trustworthiness First

Collecting outside sources that support your research and specific topic is a critical step in writing an effective paper. But before you run to the library and grab the first 20 books you can lay your hands on, keep in mind that selecting a source to include in your paper should not be taken lightly. Before you proceed with using it to backup your ideas, run a quick Internet search for it and see if other scholars in your field have written about it as well. Check to see if there are book reviews about it or peer accolades. If you spot something that seems off to you, you may want to consider leaving it out of your work. Doing this before your start making citations can save you a ton of time in the long run.

Finished with your paper? It may be time to run it through a grammar and plagiarism checker , like the one offered by EasyBib Plus. If you’re just looking to brush up on the basics, our grammar guides  are ready anytime you are.

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APA (American Psychological Association) style is most frequently used within the social sciences, in order to cite various sources. This APA Citation Guide provides the general format for in-text citations and the reference page. For more information, please consult the Publication Manual of the American Psychological Association , 7th ed.

In APA style, two citations are used to cite a source:

• A short citation used in the text (called the in-text citation ).
• A full citation (called the reference ) in the reference list at the end of a paper.

The in-text citation is a short citation that is placed next to the text being cited. The in-text citation lets the reader know that the information came from the cited source. The reference list entry provides complete details of a source and is shown at the end of a document.

In order to properly cite a source in APA style, you must have both citation types in your paper. Every in-text citation has a reference list entry. Every reference list entry has at least one (maybe more) corresponding in-text citation.

In-text citations

The basic elements needed for an in-text citation are the author’s surname and the publication year . Sometimes, page numbers are also included, especially when quotes are mentioned in the text. In-text citations are mentioned in the text in two ways: as a narrative citation or a parenthetical citation.

Narrative citations are incorporated into the text and act as a part of the sentence. Usually, narrative citations use the author’s name in the text and the publication year is enclosed in parenthesis after the name. An example of a narrative citation for one author is given below:

Barbarin (2013) examined socioemotional learning in African boys.

Parenthetical

Parenthetical citations add the author’s name and the publication year at the end of the sentence in parenthesis. An example of a parenthetical citation is given below:

Inhibition and working memory in young children were studied extensively (Aase, 2014).

When are page numbers are included?

Page numbers are referred to within in-text citations when quotes are used. Examples of both narrative citations and parenthetical citations are given below.

Ahmed (2004, p. 44)

Ahmed (2004, pp. 53–56)

Parenthetical:

(Ahmed, 2004, p. 44)

(Ahmed, 2004, pp. 53–56)

Examples of in-text citations

Here are a few examples of in-text citations for a different number of authors:

Use the surname of the author in in-text citations. Use a comma before the publication year in parenthetical citations.

Narrative: 

Bucher (2018)

Parenthetical: 

(Bucher, 2018)

Two authors

Separate the author surnames with an “and” in narrative citations. Use an ampersand symbol (&) in parenthetical citations.

Popescu and Pennacchiotti (2010)

(Popescu & Pennacchiotti, 2010)

Three or more authors

Use the first author surname name followed by et al.

van Dijck et al. (2018)

(van Dijck et al., 2018)

Group author

Treat the group author similar to how you would treat author names.

Auger Collaboration (2003)

(Auger Collaboration, 2018)

If there is no author for the source, use the source title in place of the author’s name. In general, sources with no author appear as parenthetical citations.

When you add such in-text citations, you will either italicize the text or place it in quotations. If the source title is italicized in the reference list entry, italicize the title in the in-text citation. If the title is not italicized, place it in quotation marks.

Parenthetical, book:

( Nothing here , 1997)

Parenthetical, journal article:

(“Examination of parrotfish impact on coral reefs,” 2018)

Reference list entries

Reference list entries are also called full citations. There are four main details that most reference list entries have:

• The author field.
• The publication year.
• The title of the work ( italicized or in “quotation marks”).
• The source from where the reference can be obtained (e.g., URL, DOI, etc.).

Depending on the source type, you will also need additional details like volume number, publication title, contributors, medium, etc.

Examples of reference list entries

Below are a few examples of different types of reference entries along with their templates. The examples given are for one author. Note that “F” and “M” in the templates denote the first and the middle initials of an author’s name.

The title of the book is set in italics and sentence case.

Author Surname, F. M. (Publication Year). Title of the book . Publisher.

Ahmed, S. (2014). The cultural politics of emotion . Edinburgh University Press.

Journal article

The title of the article is in sentence case. The first word of a subtitle is capitalized. The journal title and the volume number are set in italics. If an article has a DOI it should always be included. Use “https://doi.org/” before the DOI. If there is no DOI for an online journal, include the URL instead. Do not use a period after the DOI or URL.

Author Surname, F. M. (Publication Year). Article title: Subtitle. Journal Title, Volume (issue), page range. URL or DOI

Collins, R. (2004). Rituals of solidarity and security in the wake of terrorist attack. Sociological Theory, 22 (1), 53–87. https://doi.org/10.1111/j.1467-9558.2004.00204.x

Newspaper or magazine article

Newspaper and magazine articles take the same style. The title of the article is in plain text and sentence case; the title of the newspaper or the magazine is set in italics. Follow the format given in the template and example for setting the date, month, and year.

Surname, F. M. (Date of publication). Title of the article. Title of the Newspaper or Magazine . URL

TNN. (2021, July 18). Parents have a habit of comparing kids to others but you don’t need to. The Times of India . https://timesofindia.indiatimes.com//home/sunday-times/parents-have-a-habit-of-comparing-kids-to-others-but-you-dont-need-to/articleshow/84507857.cms

The webpage title is in plain text, while the Website name is set in italics. Follow the format given in the template and example for setting the date, month, year, and URL.

Author or Organization Name. (Year, Month Day of Publication ). Webpage title. Title of the Website. URL

Lamberth, H. (2021, August 12). Binge drinking is problem drinking: How to get back in control. PSYCOM . https://www.psycom.net/binge-drinking-problem-drinking

YouTube video

The video title is set in sentence case and italicized. The first word after a colon is capitalized. The word “Video” is enclosed in brackets after the video title. This is followed followed by the word “YouTube.” Finally, the link is given. Note that a period is not given after the URL.

Uploader’s name, F. (Year, Month Day Published). Video title [Video]. YouTube. URL

Ananta, P. (2021, February 21). APJ Abdul Kalam inspirational quotes [Video]. YouTube. https://www.youtube.com/watch?v=pjfL51RFL2k

Reference entries for different number of authors

The number of authors in the source decides how the author name(s) will be set in the references list. Here, you will see many journal references with different numbers of authors.

List the author name followed by the publication year.

Author Surname, F. M. (Publication Year). Article title: Subtitle. Journal Title, Volume (issue), page range.

Spitka, T. (2017). Mediating among mediators: Building a consensus in multilateral interventions. International Negotiation, 23 , 1–30.

Separate the author names by an ampersand. Use a comma between the first author’s initial and the ampersand symbol.

Author Surname, F. M., & Author Surname, F. (Publication Year). Article title: Subtitle. Journal Title, Volume (issue), page range. DOI or URL

Bernstein, B., & Solomon, J. (1999). Pedagogy, identity and the construction of a theory of symbolic control: Basil Bernstein questioned by Joseph Solomon. British Journal of Sociology of Education, 20 (2), 265–279. https://doi:10.1080/01425699995443

When you add two organizations in the author field, do not use a comma before the ampersand.

Organization 1 & Organization 2. (Publication Year). Article title: Subtitle. Journal Title, Volume (issue), page range. DOI or URL

American Psychological Association & American Psychological Society. (2020). Psychology of children. Journal of Child Psychology, 34 (23), 1–12.

3–20 authors

List all author names. Do not forget to insert an “ampersand” before the last author. The example given below is for three authors.

Author Surname, F. M., Author Surname, F. M., & Author Surname, F. M. (Publication Year). Article title: Subtitle. Journal Title, Volume (issue), page range. DOI or URL

Pyysiäinen, J., Halpin, D., & Guilfoyle, A. (2017). Neoliberal governance and ‘responsibilization’ of agents: Reassessing the mechanisms of responsibility-shift in neoliberal discursive environments. Distinktion: Journal of Social Theory, 18 (2), 215–235. https://doi:10.1080/1600910X.2017.1331858

More than 20 authors

List the names of the first 19 authors followed by an ellipsis. Add the final author name after the ellipsis but without the ampersand symbol before the last author name.

Author Surname1, F. M., Author Surname2, F. M., Author Surname3, F. M., Author Surname4, F. M., Author Surname5, F. M., Author Surname6, F. M., Author Surname7, F. M., Author Surname8, F. M., Author Surname9, F. M., Author Surname10, F. M., Author Surname11, F. M., Author Surname12, F. M., Author Surname13, F. M., Author Surname14, F. M., Author Surname15, F. M., Author Surname16, F. M., Author Surname17, F. M., Author Surname18, F. M.,  Author Surname19, F. M,¼ Last Author name, F. M. (Publication Year). Article title: Subtitle. Journal Title, Volume (issue), page range. DOI or URL

Fox, J., Harper, D., Bird, A., Kindler, F. A., Feng, H.-G., Seng, A. L., Sevel, K., Ed, E., Nell, A., Ten, T., Elin, K. J., Thomas, A., Thendy, S., Fall, W., Fint, E., Gurdy, A. K., Dondy, D., Egert, E., Nanda, A. L., ¼ Long, G.  (2015). Pedagogising knowledge: Bernstein’s theory of the pedagogic device. British Journal of Sociology of Education, 23 (4), 571–582.

For additional information on APA format, select from one of the source types below. For help creating APA citations, check out the BibMe APA citation generator.

Source Types:

• How to cite a Book in APA
• How to cite a Magazine in APA
• How to cite a Newspaper in APA
• How to cite a Website in APA
• How to cite a Journal Article in APA
• How to cite a Film in APA
• How to cite an Interview in APA
• How to cite a Lecture in APA
• How to cite a TV Show / Radio Broadcast in APA
• How to cite an Encyclopedia in APA
• How to cite a Photograph in APA
• APA 7 Updates

APA Format:

• In-Text Citation Basics
• Reference Page

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As per Section 8.17 from the Publication Manual of the American Psychological Association , for any work that has three or more authors, the name of the first author and “et al.” should be used as in-text citation. The Latin phrase “et al” means “and others” and is used to reduce the citation length.

Example In-Text Citation Entry:

No stretch of reason can categorize cultural appropriation as imaginary (Rahim et al., 2020).

Sometimes, the same set of initial authors and the same publication year appear in a paper. In such rare circumstances, as per Section 8.18 of the APA manual, write out as many names as needed to differentiate between these similar references.

Example In-Text Citation Entries:

Miller, John, Reighstag et al. (2018)

Miller, John, Amudsen, et al. (2018)

As per Section 8.21 and Table 8.1 of the APA Publication Manual , a citation for a group author may be abbreviated in in-text citations. It is not compulsory to do so; however, if the group author is well known or if it appears at least thrice in the paper, then the name of the group may be abbreviated.

Parenthetical in-text citation template and example:

(Full Name of the Group [Abbreviation], year)

(National Institute of Mental Health [NIMH], 2018)

Whether it is a narrative or parenthetical in-text citation, the full name of the group should be mentioned in the first instance, along with the abbreviation.

Narrative in-text citation examples:

The American Psychological Association (APA, 2017) argues that… (first instance)

As per the APA (2017), it is standard practice that… (subsequent instances)

APA Style (7th ed.)

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American Psychological Association (APA) Style is used by writers in the social sciences:

Business, Communications, Education, Geography, Political Science, Psychology, Sociology & Anthropology, and Women's & Gender Studies

Editable Template Documents for Student Papers

• APA 7th ed. Template Document This is an APA format template document in Google Docs. Click on the link -- it will ask for you to make a new copy of the document, which you can save in your own Google Drive with your preferred privacy settings.
• APA 7th ed. Template Document A Microsoft Word document formatted correctly according to APA 7th edition.
• APA 7th ed. Annotated Bibliography template A Microsoft Word document formatted correctly for an annotated bibliography.

Example Student Paper

• APA Example Paper An annotated sample student paper from APA.

Printable Handouts & Tutorial

• Citing References: APA Style [PDF]

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Purdue Online Writing Lab Purdue OWL® College of Liberal Arts

In-Text Citations: The Basics

Welcome to the Purdue OWL

This page is brought to you by the OWL at Purdue University. When printing this page, you must include the entire legal notice.

Copyright ©1995-2018 by The Writing Lab & The OWL at Purdue and Purdue University. All rights reserved. This material may not be published, reproduced, broadcast, rewritten, or redistributed without permission. Use of this site constitutes acceptance of our terms and conditions of fair use.

Note:  This page reflects the latest version of the APA Publication Manual (i.e., APA 7), which released in October 2019. The equivalent resource for the older APA 6 style  can be found here .

Reference citations in text are covered on pages 261-268 of the Publication Manual. What follows are some general guidelines for referring to the works of others in your essay.

Note:  On pages 117-118, the Publication Manual suggests that authors of research papers should use the past tense or present perfect tense for signal phrases that occur in the literature review and procedure descriptions (for example, Jones (1998)  found  or Jones (1998)  has found ...). Contexts other than traditionally-structured research writing may permit the simple present tense (for example, Jones (1998)  finds ).

APA Citation Basics

When using APA format, follow the author-date method of in-text citation. This means that the author's last name and the year of publication for the source should appear in the text, like, for example, (Jones, 1998). One complete reference for each source should appear in the reference list at the end of the paper.

If you are referring to an idea from another work but  NOT  directly quoting the material, or making reference to an entire book, article or other work, you only have to make reference to the author and year of publication and not the page number in your in-text reference.

On the other hand, if you are directly quoting or borrowing from another work, you should include the page number at the end of the parenthetical citation. Use the abbreviation “p.” (for one page) or “pp.” (for multiple pages) before listing the page number(s). Use an en dash for page ranges. For example, you might write (Jones, 1998, p. 199) or (Jones, 1998, pp. 199–201). This information is reiterated below.

Regardless of how they are referenced, all sources that are cited in the text must appear in the reference list at the end of the paper.

In-text citation capitalization, quotes, and italics/underlining

• Always capitalize proper nouns, including author names and initials: D. Jones.
• If you refer to the title of a source within your paper, capitalize all words that are four letters long or greater within the title of a source:  Permanence and Change . Exceptions apply to short words that are verbs, nouns, pronouns, adjectives, and adverbs:  Writing New Media ,  There Is Nothing Left to Lose .

( Note:  in your References list, only the first word of a title will be capitalized:  Writing new media .)

• When capitalizing titles, capitalize both words in a hyphenated compound word:  Natural-Born Cyborgs .
• Capitalize the first word after a dash or colon: "Defining Film Rhetoric: The Case of Hitchcock's  Vertigo ."
• If the title of the work is italicized in your reference list, italicize it and use title case capitalization in the text:  The Closing of the American Mind ;  The Wizard of Oz ;  Friends .
• If the title of the work is not italicized in your reference list, use double quotation marks and title case capitalization (even though the reference list uses sentence case): "Multimedia Narration: Constructing Possible Worlds;" "The One Where Chandler Can't Cry."

Short quotations

If you are directly quoting from a work, you will need to include the author, year of publication, and page number for the reference (preceded by "p." for a single page and “pp.” for a span of multiple pages, with the page numbers separated by an en dash).

You can introduce the quotation with a signal phrase that includes the author's last name followed by the date of publication in parentheses.

If you do not include the author’s name in the text of the sentence, place the author's last name, the year of publication, and the page number in parentheses after the quotation.

Long quotations

Place direct quotations that are 40 words or longer in a free-standing block of typewritten lines and omit quotation marks. Start the quotation on a new line, indented 1/2 inch from the left margin, i.e., in the same place you would begin a new paragraph. Type the entire quotation on the new margin, and indent the first line of any subsequent paragraph within the quotation 1/2 inch from the new margin. Maintain double-spacing throughout, but do not add an extra blank line before or after it. The parenthetical citation should come after the closing punctuation mark.

Because block quotation formatting is difficult for us to replicate in the OWL's content management system, we have simply provided a screenshot of a generic example below.

Formatting example for block quotations in APA 7 style.

Quotations from sources without pages

Direct quotations from sources that do not contain pages should not reference a page number. Instead, you may reference another logical identifying element: a paragraph, a chapter number, a section number, a table number, or something else. Older works (like religious texts) can also incorporate special location identifiers like verse numbers. In short: pick a substitute for page numbers that makes sense for your source.

Summary or paraphrase

If you are paraphrasing an idea from another work, you only have to make reference to the author and year of publication in your in-text reference and may omit the page numbers. APA guidelines, however, do encourage including a page range for a summary or paraphrase when it will help the reader find the information in a longer work. 

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A Quick Guide to Harvard Referencing | Citation Examples

Published on 14 February 2020 by Jack Caulfield . Revised on 15 September 2023.

Referencing is an important part of academic writing. It tells your readers what sources you’ve used and how to find them.

Harvard is the most common referencing style used in UK universities. In Harvard style, the author and year are cited in-text, and full details of the source are given in a reference list .

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Table of contents

Harvard in-text citation, creating a harvard reference list, harvard referencing examples, referencing sources with no author or date, frequently asked questions about harvard referencing.

A Harvard in-text citation appears in brackets beside any quotation or paraphrase of a source. It gives the last name of the author(s) and the year of publication, as well as a page number or range locating the passage referenced, if applicable:

Note that ‘p.’ is used for a single page, ‘pp.’ for multiple pages (e.g. ‘pp. 1–5’).

An in-text citation usually appears immediately after the quotation or paraphrase in question. It may also appear at the end of the relevant sentence, as long as it’s clear what it refers to.

When your sentence already mentions the name of the author, it should not be repeated in the citation:

Sources with multiple authors

When you cite a source with up to three authors, cite all authors’ names. For four or more authors, list only the first name, followed by ‘ et al. ’:

Sources with no page numbers

Some sources, such as websites , often don’t have page numbers. If the source is a short text, you can simply leave out the page number. With longer sources, you can use an alternate locator such as a subheading or paragraph number if you need to specify where to find the quote:

Multiple citations at the same point

When you need multiple citations to appear at the same point in your text – for example, when you refer to several sources with one phrase – you can present them in the same set of brackets, separated by semicolons. List them in order of publication date:

Multiple sources with the same author and date

If you cite multiple sources by the same author which were published in the same year, it’s important to distinguish between them in your citations. To do this, insert an ‘a’ after the year in the first one you reference, a ‘b’ in the second, and so on:

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A bibliography or reference list appears at the end of your text. It lists all your sources in alphabetical order by the author’s last name, giving complete information so that the reader can look them up if necessary.

The reference entry starts with the author’s last name followed by initial(s). Only the first word of the title is capitalised (as well as any proper nouns).

Sources with multiple authors in the reference list

As with in-text citations, up to three authors should be listed; when there are four or more, list only the first author followed by ‘ et al. ’:

Reference list entries vary according to source type, since different information is relevant for different sources. Formats and examples for the most commonly used source types are given below.

• Entire book
• Book chapter
• Translated book
• Edition of a book

Journal articles

• Print journal
• Online-only journal with DOI
• Online-only journal with no DOI

• General web page
• Online article or blog
• Social media post

Sometimes you won’t have all the information you need for a reference. This section covers what to do when a source lacks a publication date or named author.

No publication date

When a source doesn’t have a clear publication date – for example, a constantly updated reference source like Wikipedia or an obscure historical document which can’t be accurately dated – you can replace it with the words ‘no date’:

Note that when you do this with an online source, you should still include an access date, as in the example.

When a source lacks a clearly identified author, there’s often an appropriate corporate source – the organisation responsible for the source – whom you can credit as author instead, as in the Google and Wikipedia examples above.

When that’s not the case, you can just replace it with the title of the source in both the in-text citation and the reference list:

Harvard referencing uses an author–date system. Sources are cited by the author’s last name and the publication year in brackets. Each Harvard in-text citation corresponds to an entry in the alphabetised reference list at the end of the paper.

Vancouver referencing uses a numerical system. Sources are cited by a number in parentheses or superscript. Each number corresponds to a full reference at the end of the paper.

A Harvard in-text citation should appear in brackets every time you quote, paraphrase, or refer to information from a source.

The citation can appear immediately after the quotation or paraphrase, or at the end of the sentence. If you’re quoting, place the citation outside of the quotation marks but before any other punctuation like a comma or full stop.

In Harvard referencing, up to three author names are included in an in-text citation or reference list entry. When there are four or more authors, include only the first, followed by ‘ et al. ’

Though the terms are sometimes used interchangeably, there is a difference in meaning:

• A reference list only includes sources cited in the text – every entry corresponds to an in-text citation .
• A bibliography also includes other sources which were consulted during the research but not cited.

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If you want to cite this source, you can copy and paste the citation or click the ‘Cite this Scribbr article’ button to automatically add the citation to our free Reference Generator.

Caulfield, J. (2023, September 15). A Quick Guide to Harvard Referencing | Citation Examples. Scribbr. Retrieved 12 August 2024, from https://www.scribbr.co.uk/referencing/harvard-style/

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In-text citations

Two or more works cited at one point in the text

If two or more works by different authors or authoring bodies are cited at one point in the text, use a semi-colon to separate them:

(Larsen 2000; Malinowski 1999)

The authors should be listed in alphabetical order.

Two or three authors or authoring bodies

When citing a work by two or three authors or authoring bodies, cite the names in the order in which they appear on the title page:

(Malinowski, Miller & Gupta 1995) 

In-Text & Reference List Examples

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Harvard Guide to Using Sources 

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• When neither the author nor the page number is mentioned in the body of the sentence, you should include both the author’s last name and the page number in the parenthetical citation.

Colleges and universities need to create policies that foster inclusion for low-income students (Jack 24).

• When the author’s name is mentioned in the sentence, you should include only the page number in your parenthetical citation.

As Anthony Jack argues, colleges and universities need to create policies that foster inclusion for low-income students (24).

• If the source you are writing about does not have page numbers, or if you consulted an e-book version of the source, you should include only the author’s name in the parenthetical citation:

Colleges and universities need to create policies that foster inclusion for low-income students (Jack).

• If you mention the author in the body of the sentence and there is no page number in the source, you should not include a parenthetical citation.

As Anthony Jack argues, colleges and universities need to create policies that foster inclusion for low-income students.

• If you are referring to an entire work rather than a specific page, you do not need to include a page number.

In The Privileged Poor, Anthony Jack describes many obstacles that low-income students face at selective colleges and universities.

• If you are referring to a source that has no listed author, you should include the title (or a shortened version of the title) in your parenthetical citation.

Harvard College promises “to educate the citizens and citizen-leaders for our society” (“Mission, Vision, & History”).

• If you are referring to a source that has two authors, you should include both authors in your parenthetical citation.

The researchers tested whether an intervention during the first year of college could improve student well-being (Walton and Cohen 1448).

• If you refer to a source that has more than two authors, you should include the first author’s name followed by et al. ( Et al. is an abbreviation for et alia which means “and others” in Latin.) When you use et al. in a citation, you should not put it in italics.

The researchers studied more than 12,000 students who were interested in STEM fields (LaCosse et al. 8).

• If you refer to more than one source by the same author in your paper, you should include the title (or a shortened version of the title) in your parenthetical citation so that readers will know which source to look for in your Works Cited list. If you mention the author’s name in the sentence, you only need to include the title and page number. If you mention the author and title in the sentence, you only need to include the page number.

Colleges and universities need to create policies that foster inclusion for low-income students (Jack, Privileged Poor 24).

According to Anthony Jack, colleges and universities need to create policies that foster inclusion for low-income students ( Privileged Poor 24).

As Anthony Jack writes in Privileged Poor, colleges and universities need to create policies that foster inclusion for low-income students (24).

• If you want to credit multiple authors for making the same point, you can include them all in one parenthetical citation. 

Students who possess cultural capital, measured by proxies like involvement in literature, art, and classical music, tend to perform better in school (Bourdieu and Passeron; Dumais; Orr).

• If you refer to a source that includes line numbers in the margins, numbered paragraphs, numbered chapters, or numbered sections rather than page numbers, you should include the number in your parenthetical citation, along with “line,” “ch./ chs.,” or “sec./secs.”   You can include stable numbering like chapters even when there are no stable page numbers (as in an e-book). You should separate “line” or other designation from the work’s title or author’s name with a comma.  If the source does not include this type of numbering, you should not include it either.

We learn that when he went to the store to buy clothes for his son, “a frantic inspection of the boys’ department revealed no suits to fit the new-born Button” (Fitzgerald, ch.2).

• If you are citing a play, you should include the act and scene along with line numbers (for verse) or page numbers, followed by act and scene, (for prose).

Guildenstern tells Hamlet that “there has been much throwing about of brains” (Shakespeare, 2.2. 381-382).

Chris is in this mindset when he says, “a couple minutes, and your whole life changes, that’s it. It’s gone” (Nottage, 13; act 1, scene1).

• If you are referring to a video or audio recording that contains time stamps, you should include the time in your parenthetical citation to make it easy for your readers to find the part of the recording that you are citing.

In the Stranger Things official trailer, the audience knows that something unusual is going to happen from the moment the boys get on their bicycles to ride off into the night (0:16).

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Citing Sources: Sample Citations

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Sample Citations

• Major Changes to the New MLA
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Many more samples of citations presented in the MLA style can be found in the current edition of The MLA Handbook for Writers of Research Papers on Reserve in the library. Please consult this book or a librarian for help with unusual resources.

Book with One Author:

Note: italicize the title of the book. (Note indents, order of authors' names and use of periods.)

Gleick, James. Chaos: Making a New Science. Penguin, 1987. 

Book with Two Authors:

Gilbert, Sandra M., and Susan Gubar. The Madwoman in the Attic: The Woman Writer and the

Nineteenth-Century Literary Imagination . Yale UP, 1979. 

Chapter or Essay in a Book/ Anthology:

Bordo, Susan.  "The Moral Content of Nabokov's Lolita."  Aesthetic Subjects,  edited by Pamela R. Matthews and  David McWhirter,

U of Minnesota P, 2003, pp. 125-52. 

A Translated Book:

After the title, add: "Translated by," and follow with the name(s) of the translator(s).

Foucault, Michel. Madness and Civilization: A History of Insanity in the Age of Reason . Translated by Richard Howard, Vintage-

Random House, 1988. 

Print Scholarly Article or Article with Page Numbers:

Put the title of the article (the title of the source) in quotes and italicize the title of the journal. This sample includes the volume (136) and issue (2) numbers after the title. (In a bibliography, include indents, page numbers, and use of periods and commas.)

Blackburn, Robin. "Economic Democracy: Meaningful, Desirable, Feasile?" Daedalus,  vol. 136, no. 2, 2007, pp. 36-45. 

Scholarly Article Online Only/No Page Numbers:

Albada, Kelly F. "The Public and Private Dialogue about the American Family on Television." Journal of

Communication,  vol. 50, no. 4, Dec. 2000, pp. 79-110. Wiley-Blackwell Journals, doi: 10.1111/j.1460-2466.2000.tb02864.x.

Popular Article (with two authors):

Put the title of the article in quotes and italicize the title of the journal. (In a bibliography. Note the order of author's names, indents, page numbers, and use of periods and colons.)

Weintraub, Arlene, and Laura Cohen. "A Thousand-Year Plan for Nuclear Waste." Business Week, 6

May 2002, pp. 94-96. 

Weintraub, Arlene, and Laura Cohen. "A Thousand-Year Plan for Nuclear Waste."  Business Week,  6

May 2002, pp. 94-96. http://www.bloomberg.com/news/articles/2002-05-05/a-thousand-year-plan-for-nuclear-waste.

If accessed through a database:

Weintraub, Arlene, and Laura Cohen. "A Thousand-Year Plan for Nuclear Waste." Business Week,  6 May 2002, pp. 94-96.  Business

Source Complete,  https://dickinson.idm.oclc.org/login?url=http://search.ebscohost.com/login.aspx?

direct=true&db=bth&AN=6566717&site=eds-live&scope=site.

Newspaper Article:

If there is more than one edition available for that date:

Brubaker, Bill. "New Health Center Targets County's Uninsured Patients." Washington Post, 24 May 2007, p. LZ01. 

Krugman, Andrew. "Fear of Eating."  New York Times,   late  e d.,  21 May 2007, A1. 

If the newspaper is less well-known or local publication, names of cities not part of titles of foreign newspapers are added in brackets after the title and are not italicized in a bibliography.

Behre, Robert. "Presidential Hopefuls Get Final Crack at Core of S.C. Democrats." Post and Courier [Charleston], 29 Apr. 2007,

p. A11. 

Trembacki, Paul. "Brees Hopes to Win Heisman for Team." Purdue Exponent [West Lafayette], 5 Dec. 2000, p. 20. 

Newspaper Article accessed in a database:

If the newspaper is less well-known or local publication, names of cities not part of titles of foreign newspapers are added in brackets after the title and are not italicized in a bibliography.

Kenedy, T. "Low Unemployment Levels and a Buoyant Economy are Boon to Alberta Credit Unions."  Globe and Mail  [Toronto],

11 Oct. 1977, p. B3.  AGRIS,  https://dickinson.idm.oclc.org/login?url=http://search.ebscohost.com/login.aspx?

direct=true&db=edsagr&AN=edsagr.US201302483303&site=eds-live&scope=site.

Encyclopedia article:

If the entries are arranged alphabetically, it is not necessary to include page numbers.

If citing an online reference source,

Dickens, Peter. "Darwin, Charles (1809-1882)." Cambridge Dictionary of Sociology,  Cambridge UP, 2006.

Credo Reference,  http://envoy.dickinson.edu:2048/login?url=http://search.credoreference.com/content/entry/cupsoc/darwin_charles_1809_1882/0 .

If citing the print version,

Dickens, Peter. "Darwin, Charles (1809-1882)." Cambridge Dictionary of Sociology,  Cambridge UP, 2006,  pp. 210-133. 

If any elements of the citation are unavailable, then skip those elements.

“College Receives Grant from Pa. DEP for Watershed Protection.” Dickinson College News Releases,  Dickinson

College, 8 Apr. 2010, < http://www.dickinson.edu /story.aspx?id=10737429666 >.

Peralta, Eyder. “Crowd Gathers In New York, Ahead Of Wall Street Protest.” National Public Radio,  ​17 Nov. 2011, < http://www.npr.org/blogs/thetwo-way/2011 /11/17/142439078/

crowd-gathers-in-new-york-ahead-of-wall-street-protest >.

“Warhol: Headlines.” National Gallery of Art , 2011, < http://www.nga.gov/exhibitions/warholinfo.shtm >.

In the above examples National Public Radio and National Gallery of Art are duplicated twice because they are the title of the website (the container) and the publisher.

Published Letters:

Woolf, Virginia. " 1138:  To T.S. Eliot." 28 July 1920.  The Letters of Virginia Woolf,  edited by Nigel Nicolson and Joanne

Trautmann, vol. 2, Harcourt Brace Jovanovich, 1976, pp. 437-38. 

Unpublished Letters in Archives:

Benton, Thomas Hart. Letter to Charles Fremont. 22 June 1847. John Charles Fremont Papers, Southwest Museum Library, Los

Angeles. Manuscript.

Unpublished Correspondence:

Hatch, James C. Letter to the author. Received  by  Daniel J. Cahill.  5 Apr. 2008. 

Boyle, Anthony T. "Re: Utopia." Received by Daniel J. Cahill. 21 June 1997. 

Cahill, Daniel J. Memo to English dept. fac.,  1 June 2000.  Brooklyn Technical High School, New York.  

Advertisement:

Air Canada. Advertisement. CNN, 15 May 1998. 

The Fitness Fragrance by Ralph Lauren. Advertisement. GQ, Apr. 1997, 111-12. 

Head and Shoulders. Advertisement. Newsweek,  17 March 2008, p. 2. 

Lecture, Speech, and Address, or Reading:

Alter, Robert, and Marilynne Robinson. "The Psalms: A Reading and Conversation." 92Y: Cultural Institution and Community Center,

New York, 17 Dec. 2007. Reading.

Matuzzi, Robert. "Archive Trauma." Archive Trouble, MLA Annual Convention, Hyatt Regency, Chicago, 29 Dec. 2007. 

A Manuscript or Typescript:

Including descriptor like "Manuscript" or "Typescript" is optional and only necessary if unclear what the source is.

Chaucer, Geoffrey. The Canterbury Tales. Late fourteenth century, British Lib., London, MS Harley 7334. 

Dickinson Emily. "Distance is Not the Realm of Fox." 1870, Literary and Historical Manuscripts, Pierpont Morgan Lib., New York.

Henderson, George Wylie. Baby Lou and the Angel Bud.  Mid twentieth century. Collection of Roslyn Kirkland Allen, New York.

Typescript.

Jones, Celia. "Shakespeare's Dark Lady Illuminated." 1988. Typescript .

Blog Entries:

Blog entries should follow the same citation style as web pages. List author and date published if possible.  URL is recommended but not required- check if your professor requires it!

Nizza, Mike. “Go Ahead, Annoy Away, an Australian Court Says.” The Lede,  New York Times, 15 July 2008. 

 < http://thelede.blogs.nytimes.com/2008/07/15/ >.

Doctorow, Cory. “How the Tax Code Works for Billionaires.” Boing Boing,  28 Nov. 2011. 

< http://boingboing.net/2011/11/28/how-the-tax-code-works-for-bil.html >.

Films or Movies:

The Usual Suspects . Directed by Bryan Singer, performances by Kevin Spacey, Gabriel Byrne, Chazz Palminteri, Stephen Baldwin, and Benecio del Toro, Polygram, 1995.

To emphasize specific performers or directors, begin the citation with the name of the desired performer or director, followed by the appropriate title for that person:

Lucas, George, director. Star Wars Episode IV: A New Hope . Twentieth Century Fox, 1977.

For image reproductions, treat the book or website as a container, so  the creator of the image would be the author and the title of the image would be the title.  Remember that for a second container, the title is listed first, before the contributors. For specific works of art, see the section. Cite following the guidelines for the type of resource, but make a note of where the visual is included in the resource, ie page or figure number. Cite the creator of the image as the author and the title of the image instead of an article title.

For a more detailed guide on citing images, go to Citing Images in MLA Style .

Work of Visual Art including Photographs in a Museum Collection:

MLA does not require you to include the dimensions, but it is safer to include it when citing for art and art history. 

Artist.  Title of Work . Date of Composition, Medium, Dimensions, Institution housing artwork, City where located (if not mentioned in the collection).​

Goya, Francisco.  The Family of Charles IV . 1800, oil on canvas, Museo del Prado, Madrid.

Image in an Article:

Shi Lu.  Girl in a Forest. 1955, Private collection, Estate of Shi Lu.   in " Confucius and Tolstoy in India: Shi Lu’s Paintings of 1970 and

the Socialist Culture of Maoist-Period China . "  Art History ,  Early View, June  2016 , pp.1-32.  Wiley-Blackwell

Journals,  doi:  10.1111/1467-8365.12217.

Online Image:

If citing an image found using Google images, cite the original source – not Google.   Access date is optional - check if your professor requires it!

Estrin, James.  A Worshiper at St. Patrick's Cathedral on Fifth Avenue in Manhattan . In "New Translation of Catholic Mass Makes Its

Debut,"  New York  Times,   New York Times,  2 7 Nov. 2011.   < http://www.nytimes.com/2011/11 /28/nyregion/for-catholics-

the-word-was-a-bit-different-amen.html?ref=us >.

Image or Photograph from a Book :

Goya, Francisco.  The Family of Charles IV . 1800, Museo del Prado, Madrid.  Gardener's Art Through the Ages , 10th ed., by Richard 

G.Tansey and Fred S. Kleiner, Harcourt Brace, p. 939.

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How to Write a Research Proposal: (with Examples & Templates)

Table of Contents

Before conducting a study, a research proposal should be created that outlines researchers’ plans and methodology and is submitted to the concerned evaluating organization or person. Creating a research proposal is an important step to ensure that researchers are on track and are moving forward as intended. A research proposal can be defined as a detailed plan or blueprint for the proposed research that you intend to undertake. It provides readers with a snapshot of your project by describing what you will investigate, why it is needed, and how you will conduct the research.  

Your research proposal should aim to explain to the readers why your research is relevant and original, that you understand the context and current scenario in the field, have the appropriate resources to conduct the research, and that the research is feasible given the usual constraints.  

This article will describe in detail the purpose and typical structure of a research proposal , along with examples and templates to help you ace this step in your research journey.  

What is a Research Proposal ?  

A research proposal¹ ,²  can be defined as a formal report that describes your proposed research, its objectives, methodology, implications, and other important details. Research proposals are the framework of your research and are used to obtain approvals or grants to conduct the study from various committees or organizations. Consequently, research proposals should convince readers of your study’s credibility, accuracy, achievability, practicality, and reproducibility.   

With research proposals , researchers usually aim to persuade the readers, funding agencies, educational institutions, and supervisors to approve the proposal. To achieve this, the report should be well structured with the objectives written in clear, understandable language devoid of jargon. A well-organized research proposal conveys to the readers or evaluators that the writer has thought out the research plan meticulously and has the resources to ensure timely completion.  

Purpose of Research Proposals  

A research proposal is a sales pitch and therefore should be detailed enough to convince your readers, who could be supervisors, ethics committees, universities, etc., that what you’re proposing has merit and is feasible . Research proposals can help students discuss their dissertation with their faculty or fulfill course requirements and also help researchers obtain funding. A well-structured proposal instills confidence among readers about your ability to conduct and complete the study as proposed.  

Research proposals can be written for several reasons:³  

• To describe the importance of research in the specific topic  
• Address any potential challenges you may encounter  
• Showcase knowledge in the field and your ability to conduct a study  
• Apply for a role at a research institute  
• Convince a research supervisor or university that your research can satisfy the requirements of a degree program  
• Highlight the importance of your research to organizations that may sponsor your project  
• Identify implications of your project and how it can benefit the audience  

What Goes in a Research Proposal?    

Research proposals should aim to answer the three basic questions—what, why, and how.  

The What question should be answered by describing the specific subject being researched. It should typically include the objectives, the cohort details, and the location or setting.  

The Why question should be answered by describing the existing scenario of the subject, listing unanswered questions, identifying gaps in the existing research, and describing how your study can address these gaps, along with the implications and significance.  

The How question should be answered by describing the proposed research methodology, data analysis tools expected to be used, and other details to describe your proposed methodology.   

Research Proposal Example  

Here is a research proposal sample template (with examples) from the University of Rochester Medical Center. 4 The sections in all research proposals are essentially the same although different terminology and other specific sections may be used depending on the subject.  

Structure of a Research Proposal  

If you want to know how to make a research proposal impactful, include the following components:¹  

1. Introduction  

This section provides a background of the study, including the research topic, what is already known about it and the gaps, and the significance of the proposed research.  

2. Literature review  

This section contains descriptions of all the previous relevant studies pertaining to the research topic. Every study cited should be described in a few sentences, starting with the general studies to the more specific ones. This section builds on the understanding gained by readers in the Introduction section and supports it by citing relevant prior literature, indicating to readers that you have thoroughly researched your subject.  

3. Objectives  

Once the background and gaps in the research topic have been established, authors must now state the aims of the research clearly. Hypotheses should be mentioned here. This section further helps readers understand what your study’s specific goals are.  

4. Research design and methodology  

Here, authors should clearly describe the methods they intend to use to achieve their proposed objectives. Important components of this section include the population and sample size, data collection and analysis methods and duration, statistical analysis software, measures to avoid bias (randomization, blinding), etc.  

5. Ethical considerations  

This refers to the protection of participants’ rights, such as the right to privacy, right to confidentiality, etc. Researchers need to obtain informed consent and institutional review approval by the required authorities and mention this clearly for transparency.  

6. Budget/funding  

Researchers should prepare their budget and include all expected expenditures. An additional allowance for contingencies such as delays should also be factored in.  

7. Appendices  

This section typically includes information that supports the research proposal and may include informed consent forms, questionnaires, participant information, measurement tools, etc.  

8. Citations  

Important Tips for Writing a Research Proposal  

Writing a research proposal begins much before the actual task of writing. Planning the research proposal structure and content is an important stage, which if done efficiently, can help you seamlessly transition into the writing stage. 3,5  

The Planning Stage  

• Manage your time efficiently. Plan to have the draft version ready at least two weeks before your deadline and the final version at least two to three days before the deadline.
• What is the primary objective of your research?  
• Will your research address any existing gap?  
• What is the impact of your proposed research?  
• Do people outside your field find your research applicable in other areas?  
• If your research is unsuccessful, would there still be other useful research outcomes?  

  The Writing Stage  

• Create an outline with main section headings that are typically used.  
• Focus only on writing and getting your points across without worrying about the format of the research proposal , grammar, punctuation, etc. These can be fixed during the subsequent passes. Add details to each section heading you created in the beginning.   
• Ensure your sentences are concise and use plain language. A research proposal usually contains about 2,000 to 4,000 words or four to seven pages.  
• Don’t use too many technical terms and abbreviations assuming that the readers would know them. Define the abbreviations and technical terms.  
• Ensure that the entire content is readable. Avoid using long paragraphs because they affect the continuity in reading. Break them into shorter paragraphs and introduce some white space for readability.  
• Focus on only the major research issues and cite sources accordingly. Don’t include generic information or their sources in the literature review.  
• Proofread your final document to ensure there are no grammatical errors so readers can enjoy a seamless, uninterrupted read.  
• Use academic, scholarly language because it brings formality into a document.  
• Ensure that your title is created using the keywords in the document and is neither too long and specific nor too short and general.  
• Cite all sources appropriately to avoid plagiarism.  
• Make sure that you follow guidelines, if provided. This includes rules as simple as using a specific font or a hyphen or en dash between numerical ranges.  
• Ensure that you’ve answered all questions requested by the evaluating authority.  

Key Takeaways   

Here’s a summary of the main points about research proposals discussed in the previous sections:  

• A research proposal is a document that outlines the details of a proposed study and is created by researchers to submit to evaluators who could be research institutions, universities, faculty, etc.  
• Research proposals are usually about 2,000-4,000 words long, but this depends on the evaluating authority’s guidelines.  
• A good research proposal ensures that you’ve done your background research and assessed the feasibility of the research.  
• Research proposals have the following main sections—introduction, literature review, objectives, methodology, ethical considerations, and budget.  

Frequently Asked Questions  

Q1. How is a research proposal evaluated?  

A1. In general, most evaluators, including universities, broadly use the following criteria to evaluate research proposals . 6  

• Significance —Does the research address any important subject or issue, which may or may not be specific to the evaluator or university?  
• Content and design —Is the proposed methodology appropriate to answer the research question? Are the objectives clear and well aligned with the proposed methodology?  
• Sample size and selection —Is the target population or cohort size clearly mentioned? Is the sampling process used to select participants randomized, appropriate, and free of bias?  
• Timing —Are the proposed data collection dates mentioned clearly? Is the project feasible given the specified resources and timeline?  
• Data management and dissemination —Who will have access to the data? What is the plan for data analysis?  

Q2. What is the difference between the Introduction and Literature Review sections in a research proposal ?  

A2. The Introduction or Background section in a research proposal sets the context of the study by describing the current scenario of the subject and identifying the gaps and need for the research. A Literature Review, on the other hand, provides references to all prior relevant literature to help corroborate the gaps identified and the research need.  

Q3. How long should a research proposal be?  

A3. Research proposal lengths vary with the evaluating authority like universities or committees and also the subject. Here’s a table that lists the typical research proposal lengths for a few universities.  

Q4. What are the common mistakes to avoid in a research proposal ?  

A4. Here are a few common mistakes that you must avoid while writing a research proposal . 7  

• No clear objectives: Objectives should be clear, specific, and measurable for the easy understanding among readers.  
• Incomplete or unconvincing background research: Background research usually includes a review of the current scenario of the particular industry and also a review of the previous literature on the subject. This helps readers understand your reasons for undertaking this research because you identified gaps in the existing research.  
• Overlooking project feasibility: The project scope and estimates should be realistic considering the resources and time available.   
• Neglecting the impact and significance of the study: In a research proposal , readers and evaluators look for the implications or significance of your research and how it contributes to the existing research. This information should always be included.  
• Unstructured format of a research proposal : A well-structured document gives confidence to evaluators that you have read the guidelines carefully and are well organized in your approach, consequently affirming that you will be able to undertake the research as mentioned in your proposal.  
• Ineffective writing style: The language used should be formal and grammatically correct. If required, editors could be consulted, including AI-based tools such as Paperpal , to refine the research proposal structure and language.  

Thus, a research proposal is an essential document that can help you promote your research and secure funds and grants for conducting your research. Consequently, it should be well written in clear language and include all essential details to convince the evaluators of your ability to conduct the research as proposed.  

This article has described all the important components of a research proposal and has also provided tips to improve your writing style. We hope all these tips will help you write a well-structured research proposal to ensure receipt of grants or any other purpose.  

References  

• Sudheesh K, Duggappa DR, Nethra SS. How to write a research proposal? Indian J Anaesth. 2016;60(9):631-634. Accessed July 15, 2024. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5037942/  
• Writing research proposals. Harvard College Office of Undergraduate Research and Fellowships. Harvard University. Accessed July 14, 2024. https://uraf.harvard.edu/apply-opportunities/app-components/essays/research-proposals  
• What is a research proposal? Plus how to write one. Indeed website. Accessed July 17, 2024. https://www.indeed.com/career-advice/career-development/research-proposal  
• Research proposal template. University of Rochester Medical Center. Accessed July 16, 2024. https://www.urmc.rochester.edu/MediaLibraries/URMCMedia/pediatrics/research/documents/Research-proposal-Template.pdf  
• Tips for successful proposal writing. Johns Hopkins University. Accessed July 17, 2024. https://research.jhu.edu/wp-content/uploads/2018/09/Tips-for-Successful-Proposal-Writing.pdf  
• Formal review of research proposals. Cornell University. Accessed July 18, 2024. https://irp.dpb.cornell.edu/surveys/survey-assessment-review-group/research-proposals  
• 7 Mistakes you must avoid in your research proposal. Aveksana (via LinkedIn). Accessed July 17, 2024. https://www.linkedin.com/pulse/7-mistakes-you-must-avoid-your-research-proposal-aveksana-cmtwf/  

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Get accurate academic translations, rewriting support, grammar checks, vocabulary suggestions, and generative AI assistance that delivers human precision at machine speed. Try for free or upgrade to Paperpal Prime starting at US$19 a month to access premium features, including consistency, plagiarism, and 30+ submission readiness checks to help you succeed.  

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How to Write Your Research Paper in APA Format

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Title Page Setup

A title page is required for all APA Style papers. There are both student and professional versions of the title page. Students should use the student version of the title page unless their instructor or institution has requested they use the professional version. APA provides a student title page guide (PDF, 199KB) to assist students in creating their title pages.

Student title page

The student title page includes the paper title, author names (the byline), author affiliation, course number and name for which the paper is being submitted, instructor name, assignment due date, and page number, as shown in this example.

Title page setup is covered in the seventh edition APA Style manuals in the Publication Manual Section 2.3 and the Concise Guide Section 1.6

Related handouts

• Student Title Page Guide (PDF, 263KB)
• Student Paper Setup Guide (PDF, 3MB)

Student papers do not include a running head unless requested by the instructor or institution.

Follow the guidelines described next to format each element of the student title page.

Professional title page

The professional title page includes the paper title, author names (the byline), author affiliation(s), author note, running head, and page number, as shown in the following example.

Follow the guidelines described next to format each element of the professional title page.

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How to Write a Bibliography [Tips with Examples]

Going through the process of writing a report as a student often feels like being thrown into the deep end without a map. While you're typically given a sample report to follow, it rarely provides the systematic guidance needed to tackle the task effectively. This lack of structured support can leave you feeling confused and unsure where to begin.

The same challenge extends to writing bibliographies. Without clear instructions, understanding how to compile and format references can be daunting. In this guide, I aim to demystify the art of how to write a bibliography, offering step-by-step instructions and additional tips to help you navigate this essential aspect of academic writing with confidence.

What is a Bibliography?

A bibliography serves as a crucial component of academic writing, which includes a detailed list of all sources consulted during research. It not only validates the credibility of your work but also aids readers in retracing your steps to verify facts, data, and insights you've presented. Here’s more on what’s the purpose of bibliographies and what does it include:

Purpose of a Bibliography:

A bibliography supports academic integrity by:

Demonstrating thorough research conducted for the assignment.

Crediting original authors for their contributions.

Enabling readers to locate and explore cited sources independently.

Providing a foundation for future scholars to build upon your research.

Components of a Bibliography:

Citation Details: Includes the author's name, title of the work, publication details (like publisher and year), and specific pages or chapters referenced.

Formatting: Follows specific style guides (e.g., MLA, APA, Chicago Manual of Style) for consistency and clarity.

Organizational Structure: Typically arranged alphabetically by author’s last name or chronologically for historical works.

Annotations (in annotated bibliographies): Brief evaluations summarizing each source’s relevance, scope, and potential bias.

Categories of Bibliographies:

Enumerative Bibliography: Lists sources categorically, such as by author or topic, without additional commentary.

Analytical Bibliography: Explores the physical attributes and evolution of a work, including details like publication history and format changes.

Annotated Bibliography: Provides annotations explaining each source's significance, aiding in understanding its relevance to the research topic.

Subject Bibliography: Organizes sources by subject matter, facilitating research within specific fields or disciplines.

National Bibliography: Compiles works published within a specific country or region, often including cultural or historical contexts.

How to Write a Bibliography Step by Step?

Learning how to write a bibliography is a crucial skill in academics. It's all about giving credit where it's due - acknowledging the sources you've used in your research. Whether you're crafting an essay bibliography or a reference list for a longer paper, the basics remain the same.

In this section, we'll explore these core elements of bibliography writing. By understanding these basics, you'll be equipped to create accurate citations regardless of the specific format required. A well-crafted bibliography not only prevents plagiarism but also demonstrates the depth of your research. So, let's have a look at the basics of how to write a bibliography for essays or research papers.

Step 1: Gather Your Resources

First things first, let's round up all the materials you've used for your research. This includes:

Books: Whether they're physical copies or e-books, make sure you've got them all listed.

Articles: This covers journal articles, magazine pieces, and newspaper reports.

Websites: Any online sources you've referenced should be included.

Other media: Don't forget about videos, podcasts, or interviews you might have used.

Maintain an ongoing list of sources as you research—it'll simplify your work later on!

Step 2: Record Citation Information

Now, here's where the real work begins. For each source, you'll need to jot down:

Author(s): Full names, please!

Title of the work: Whether it's a book title, article name, or website header.

Publication date: When was this information made available?

Publisher: Who put this information out there?

Page numbers: If you're using a physical book or a PDF with page numbers.

URL and access date: For online sources, note when you accessed the information.

Here's a quick example of what bibliographies for different sources might look like:

For a book:

Author: Jane Smith

Title: The Art of Bibliography Writing

Publication Date: 2022

Publisher: Academic Press

Pages:56-58

For a website:

Author: John Doe

Title: "10 Tips for Perfect Citations"

Publication Date: March 15, 2023

URL: www.citationtips.com

Accessed: July 17, 2024

Step 3: Format Your Bibliography

Next, we need to consider the formatting requirements. Remember, each academic style may have different formatting demands, but they all adhere to the same general rules:

Alphabetical order:

Arrange entries by the author's last name.

For works without an author, use the title (ignoring articles like "A," "An," or "The" at the beginning).

Hanging indent :

Set a hanging indent of 0.5 inches (1.27 cm) for each entry.

The first line of each entry should be flush left, with subsequent lines indented.

Double-space the entire bibliography for most styles.

Some styles may require single spacing within entries and double spacing between them.

Consistency:

Use the same punctuation, capitalization, and formatting throughout.

Pay attention to details like italicization and quotation marks.

Speaking of styles, there are several to choose from, and the one you use often depends on your field of study or your instructor's preference. The most common are:

APA (American Psychological Association)

MLA (Modern Language Association)

Don't worry – we'll look into each of these academic styles in the upcoming sections where we'll discuss the formatting requirements for each style in detail, along with a few examples to help you understand how to format a bibliography according to different academic styles!

Bibliographies in any format often involve formatting risks, especially when converting to PDF. That's why I'll demonstrate WPS Office to you, where not only will my formatting remain preserved, but it's also the ideal office suite for students. It offers free capabilities and is easily navigable

How to Write a Bibliography in APA Format?

APA (American Psychological Association) format is widely used in social sciences. Before we dive into examples, let's review the key formatting requirements:

APA Formatting Requirements:

Title the page "References" centered at the top.

Double-space all entries.

Use a hanging indent for each entry (first line flush left, subsequent lines indented 0.5 inches).

Alphabetize entries by the author's last name.

For multiple works by the same author, order chronologically from earliest to most recent.

Use only the initials for authors' first and middle names.

Now, let's look at how to cite different types of sources:

Author's last name, first initial. (Publication date). Book title. Additional information. City of publication: Publishing company.

Gladwell, M. (2008). Outliers: The story of success. Little, Brown and Company.

Journal Articles:

Author's Last Name, First Initial. Middle Initial. (Year). Title of article. Name of Journal, Volume(Issue), Page range. DOI if available

Krueger, R. F., & Markon, K. E. (2006). Reinterpreting comorbidity: A model-based approach to understanding and classifying psychopathology. Annual Review of Clinical Psychology, 2, 111-133. https://doi.org/10.1146/annurev.clinpsy.2.022305.095213

Author's Last Name, First Initial. Middle Initial. (Year, Month Day). Title of page. Website Name. URL

Centers for Disease Control and Prevention. (2023, May 5). COVID-19 vaccination clinical and professional resources. https://www.cdc.gov/vaccines/covid-19/index.html

How to Write a Bibliography in MLA Format?

MLA (Modern Language Association) format is commonly used in humanities. Let's review the formatting requirements:

Formatting Requirements:

Title the page "Works Cited" centered at the top.

If no author is given, alphabetize by the title, ignoring articles (A, An, The).

Use the full first name of authors, not just initials.

Now, these are some ways on how to cite different types of sources:

Author's Last Name, First Name. Title of Book. Publisher, Year of Publication.

Austen, Jane. Pride and Prejudice. Penguin Classics, 2002.

Author's Last Name, First Name. "Title of Article." Name of Journal, vol. Volume Number, no. Issue Number, Year of Publication, pp. Page Range.

Poe, Edgar Allan. "The Philosophy of Composition." Graham's Magazine, vol. 28, no. 4, 1846, pp. 163-167.

Author's Last Name, First Name. "Title of Web Page." Name of Website, Date of Publication or Last Update, URL. Accessed Day Month Year.

Hollmichel, Stefanie. "The Reading Brain: Differences between Digital and Print." So Many Books, 25 Apr. 2013, somanybooksblog.com/2013/04/25/the-reading-brain-differences-between-digital-and-print/. Accessed 4 May 2023.

How to Write a Bibliography in Chicago Format?

Chicago style has two systems: notes and bibliography (used in humanities) and author-date (used in sciences and social sciences). We'll focus on the notes and bibliography system. First, the formatting requirements:

Title the page "Bibliography" centered at the top.

Single-space each entry, with a blank line between entries.

Here’s how you can cite different types of sources in Chicago format:

Last Name, First Name. Title of Book. Place of Publication: Publisher, Year of Publication.

Hemingway, Ernest. The Old Man and the Sea. New York: Scribner, 1952.

Last Name, First Name. "Title of Article." Name of Journal Volume Number, no. Issue Number (Year of Publication): Page Range.

Sontag, Susan. "Against Interpretation." Evergreen Review 34 (1964): 76-84.

Last Name, First Name. "Title of Web Page." Name of Website. Publishing Organization, Publication or Modified Date. URL.

Kenzie, Susan. "The Elements of Chicago Style." Writing Resources. University of Chicago, last modified March 23, 2022. https://writingresources.uchicago.edu/chicago-style-elements .

How to Write a Bibliography in Harvard Format?

Harvard referencing style is commonly used in the UK and Australia. Let's review the formatting requirements:

Title the page "Reference List" or "References" centered at the top.

You can site different type of sources in the following ways in Harvard format:

Author's Last name, Initial(s). (Year) Title of book. Edition (if not first edition). Place of publication: Publisher.

Hawking, S. (1988) A brief history of time. London: Bantam Books.

Author's Last name, Initial(s). (Year) 'Title of article', Journal Name, Volume(Issue), Page range.

Boughton, J.M. (2002) 'The Bretton Woods proposal: an in-depth look', Political Science Quarterly, 42(6), pp. 564-578.

Author's Last name, Initial(s). (Year) Title of web page. Available at: URL (Accessed: Day Month Year).

BBC News (2023) Climate change: Scientists warn of 'irreversible' impacts. Available at: https://www.bbc.com/news/science-environment-51742646 (Accessed: 17 July 2023).

Bonus Tips: Using WPS to Perfect your Bibliography

WPS Office is a golden ticket for academic success, especially for students navigating the intricacies of bibliography writing. Often, students struggle with formatting details, but WPS AI steps in as a reliable assistant, ensuring seamless document preparation.

1.Convert Word to PDF without Losing Format

One of WPS Office's standout features is its ability to convert Word documents to PDF flawlessly. Unlike Microsoft Word, which can sometimes disrupt formatting, WPS Office ensures that your meticulously formatted bibliographies in APA, MLA, or Chicago style are preserved without any loss of structure or design. This reliability is crucial as it saves students valuable time and effort, allowing them to focus on the content rather than technical adjustments.

2.Check the Format & Spelling

WPS AI goes beyond formatting; it also checks for spelling and grammar errors with precision. This feature guarantees that your bibliographies are not only visually polished but also linguistically flawless. Students can rest assured that their academic work meets the highest standards of clarity and correctness, enhancing both their academic credibility and professional presentation skills.

1. What is annotated bibliography?

An annotated bibliography is a collection of sources that provides a summary of research related to a specific topic. It includes a list of citations for each source, accompanied by a brief descriptive text (an annotation) that summarizes and evaluates the content. This annotation helps readers understand the source's relevance and usefulness. An annotated bibliography can function as a standalone assignment or be incorporated as part of a larger research project.

2. How can I cite without author name or date?

If the source lacks an author or date, incorporate the title into your signal phrase or within parentheses, using the abbreviation "n.d." (for "no date"). Another study focusing on students and their research choices found that those who received tutoring achieved success ("Tutoring and APA," n.d.).

Format Your Academic Research With WPS Office

Your research likely consumed a great deal of time, effort, and especially involved extensive digging into research papers, books, and other materials. And you took your time learning how to write a bibliography but to ensure your research is solid and authentic, it's crucial to include all these sources in the correct format on your bibliography page. Otherwise, your research may lack credibility. WPS Office helps restore that confidence by assisting you in formatting it correctly. Download WPS Office and discover how it can significantly aid you as a student.

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As a student, am I allowed to use a Gen-AI tool like ZotGPT in my coursework?

The UCI Policy on Academic Integrity for students states:

“All students are expected to complete a course in compliance with the Instructor's standards. No student shall engage in any activity involving any Academic Integrity Policy Violations. No student shall engage in any activity that involves attempting to receive a grade by means other than honest effort, and shall not aid another student who is attempting to do so.”

Before you use any AI tool, ask yourself:

• Is using this tool in compliance with the course instructor’s standards?
• Does using this tool undermine your “honest effort” in this course to receive a grade? 

If you have any doubts, it is best to ask your instructor, and keep in mind that each course may have different standards of what might be acceptable.

If you are permitted to use an AI tool in your work and you have made the decision to use one, it may be a good idea to acknowledge your use as follows:

• Which tool(s) did you use?
• How did you use the tool (for what purpose, in what sections, etc.)?
• How did you prompt the tool?
• When did you use the tool?

At this time, citing Generative AI is an evolving conversation, and some style guides have released official guidance on how to cite AI-generated text or images, while others have not. In addition, style guides have differing guidelines of whether tools (or companies) are “authors.”

Here are some examples of differences of how Gen-AI information might be cited in different style guides:

No official guidelines have been released, but the following blog post offers suggestions : “If you’ve used ChatGPT or other AI tools in your research, describe how you used the tool in your Method section or in a comparable section of your paper. For literature reviews or other types of essays or response or reaction papers, you might describe how you used the tool in your introduction. In your text, provide the prompt you used and then any portion of the relevant text that was generated in response.”

For example:

When prompted with “Is the left brain right brain divide real or a metaphor?” the ChatGPT-generated text indicated that although the two brain hemispheres are somewhat specialized, “the notation that people can be characterized as ‘left-brained’ or ‘right-brained’ is considered to be an oversimplification and a popular myth” (OpenAI, 2023).

OpenAI. (2023). ChatGPT (Mar 14 version) [Large language model]. https://chat.openai.com/chat

The Chicago Style FAQ suggests that “conversations” with Gen-AI chatbots can be treated in a similar fashion to private emails or phone conversations that couldn’t be accessed by others. It recommends including information about how the AI tool was prompted, and suggests “you must credit ChatGPT when you reproduce its words within your own work, but unless you include a publicly available URL, that information should be put in the text or in a note–not in a bibliography or reference list.”

For example, if a prompt was not given in-text:

1. ChatGPT, response to “Explain how to make pizza dough from common household ingredients,” OpenAI, March 7, 2023.

The IEEE Editorial Style Manual for Authors (2023) does not offer any official guidance on citing AI-generated content. Some suggest treating AI-generated text similarly to private communication.

The MLA style guide uses a template of core citation elements that are flexible enough to accommodate citing AI-generated text or images. While MLA recommends that you do not treat an AI tool as an author, it suggests you should:

• cite a generative AI tool whenever you paraphrase, quote, or incorporate into your own work any content (whether text, image, data, or other) that was created by it 
• acknowledge all functional uses of the tool (like editing your prose or translating words) in a note, your text, or another suitable location 
• take care to vet the secondary sources it cites

An example of a works cited entry for a paraphrase of text output by ChatGPT might look like this:

“Describe the symbolism of the green light in the book The Great Gatsby by F. Scott Fitzgerald” prompt. ChatGPT, 13 Feb. version, OpenAI, 8 Mar. 2023, chat.openai.com/chat.

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• The Basics of In-Text Citation | APA & MLA Examples

The Basics of In-Text Citation | APA & MLA Examples

Published on March 14, 2022 by Jack Caulfield . Revised on February 28, 2024.

An in-text citation is a short acknowledgement you include whenever you quote or take information from a source in academic writing. It points the reader to the source so they can see where you got your information.

In-text citations most commonly take the form of short parenthetical statements indicating the author and publication year of the source, as well as the page number if relevant.

We also offer a free citation generator and in-depth guides to the main citation styles.

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Table of contents

What are in-text citations for, when do you need an in-text citation, types of in-text citation, frequently asked questions about in-text citations.

The point of an in-text citation is to show your reader where your information comes from. Including citations:

• Avoids plagiarism by acknowledging the original author’s contribution
• Allows readers to verify your claims and do follow-up research
• Shows you are engaging with the literature of your field

Academic writing is seen as an ongoing conversation among scholars, both within and between fields of study. Showing exactly how your own research draws on and interacts with existing sources is essential to keeping this conversation going.

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An in-text citation should be included whenever you quote or paraphrase a source in your text.

Quoting means including the original author’s words directly in your text, usually introduced by a signal phrase . Quotes should always be cited (and indicated with quotation marks), and you should include a page number indicating where in the source the quote can be found.

Paraphrasing means putting information from a source into your own words. In-text citations are just as important here as with quotes, to avoid the impression you’re taking credit for someone else’s ideas. Include page numbers where possible, to show where the information can be found.

However, to avoid over-citation, bear in mind that some information is considered common knowledge and doesn’t need to be cited. For example, you don’t need a citation to prove that Paris is the capital city of France, and including one would be distracting.

Different types of in-text citation are used in different citation styles . They always direct the reader to a reference list giving more complete information on each source.

Author-date citations (used in APA , Harvard , and Chicago author-date ) include the author’s last name, the year of publication, and a page number when available. Author-page citations (used in MLA ) are the same except that the year is not included.

Both types are divided into parenthetical and narrative citations. In a parenthetical citation , the author’s name appears in parentheses along with the rest of the information. In a narrative citation , the author’s name appears as part of your sentence, not in parentheses.

Note: Footnote citations like those used in Chicago notes and bibliography are sometimes also referred to as in-text citations, but the citation itself appears in a note separate from the text.

An in-text citation is an acknowledgement you include in your text whenever you quote or paraphrase a source. It usually gives the author’s last name, the year of publication, and the page number of the relevant text. In-text citations allow the reader to look up the full source information in your reference list and see your sources for themselves.

At college level, you must properly cite your sources in all essays , research papers , and other academic texts (except exams and in-class exercises).

Add a citation whenever you quote , paraphrase , or summarize information or ideas from a source. You should also give full source details in a bibliography or reference list at the end of your text.

The exact format of your citations depends on which citation style you are instructed to use. The most common styles are APA , MLA , and Chicago .

Check if your university or course guidelines specify which citation style to use. If the choice is left up to you, consider which style is most commonly used in your field.

• APA Style is the most popular citation style, widely used in the social and behavioral sciences.
• MLA style is the second most popular, used mainly in the humanities.
• Chicago notes and bibliography style is also popular in the humanities, especially history.
• Chicago author-date style tends to be used in the sciences.

Other more specialized styles exist for certain fields, such as Bluebook and OSCOLA for law.

The most important thing is to choose one style and use it consistently throughout your text.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

Caulfield, J. (2024, February 28). The Basics of In-Text Citation | APA & MLA Examples. Scribbr. Retrieved August 15, 2024, from https://www.scribbr.com/citing-sources/in-text-citation-styles/

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Bayesian Machine Learning: Function Space

Learning And Generalization

Aug. 13, 2024

*Denotes Equal Contribution

S. Prince (08/1). Bayesian Machine Learning: Function Space. Borealis AI. https://www.borealisai.com/research-blogs/bayesian-machine-learning-function-space/

This blog is the fifth in a series in which we consider machine learning from four different viewpoints. We either use gradient descent or a fully Bayesian approach and for each, we can choose to focus on either the model parameters or the output function (figure 1). In part I , part II , and part III , we considered gradient descent in both parameter and function space; this led to the development of the neural tangent kernel.

Figure 1. Four approaches to model fitting. We can either consider gradient descent (top row) or a Bayesian approach (bottom row). For either, we can consider either parameter space (left column) or the function space (right column). This blog will consider the Bayesian approach in function space (lower-right panel). Figure inspired by Sohl-Dickstein (2021).

In part IV of the series we considered the Bayesian approach to model fitting in a simple linear regression model from the perspective of parameter space. Here, we determine a probability distribution over possible parameter values rather than choosing a single “best” estimate. To make a prediction, we take a weighted average the predictions from all possible parameter estimates, where the weights are based on these probabilities. This scheme has the merits of (i) making more moderate predictions and (ii) providing estimates of uncertainty over these predictions.

In this (part V) of this series, we investigate the Bayesian approach from the perspective of function space. This is considerably simpler than the parameter space view and, for linear regression, results in the same predictions. However, as we shall see in the subsequent parts of this series, the implications of the two viewpoints are quite different for neural network models.

This blog is accompanied by a CoLab Python Notebook .

Function Space

In the function space view, we convert the prior $Pr(\boldsymbol\phi)$ over the model parameters $\boldsymbol\phi$ space into a prior $\operatorname{Pr}(\mathbf{f})$ over the function $\textbf{f}[\bullet]$ itself. This allows us to define a joint probability distribution between the function values for the training data and that for the test data. We then compute the conditional distribution of the function value for the test data given the training data.

Linear Model

These ideas are easiest to understand with the concrete example of the linear model:

\begin{align*} y & =\mathrm{f}[\mathbf{x}, \boldsymbol{\phi}] \\ & =\mathbf{x}^{T} \boldsymbol{\phi}, \tag{1} \end{align*}

where $\mathbf{x} \in \mathbb{R}^{D}$ is the input data, $y \in \mathbb{R}$ is the model output and $\boldsymbol{\phi} \in \mathbb{R}^{D}$ contains the parameters. As usual, we append the value one to the input data so that $\mathbf{x} \leftarrow$ $\left[\mathbf{x}^{T}, 1\right]^{T}$ to account for the y-offset.

As before, we’ll concatenate all of the $I$ training inputs $\mathbf{x}_{i} \in \mathbb{R}^{D}$ into a single matrix $\mathbf{X} \in \mathbb{R}^{D \times I}$, all of the function outputs $\mathrm{f}[\mathbf{x}, \boldsymbol{\phi}] \in \mathbb{R}$ into a single column vector $\mathbf{f} \in \mathbb{R}^{D}$ and all of the training outputs $y_{i} \in \mathbb{R}$ into a single column vector $\mathbf{y} \in$ $\mathbb{R}^{D}$. We can then write the model in two stages:

\begin{align*} \mathbf{f} & =\mathbf{X}^{T} \boldsymbol{\phi} \\ \operatorname{Pr}(\mathbf{y} \mid \mathbf{f}) & =\operatorname{Norm}_{\mathbf{y}}\left[\mathbf{f}, \sigma_{n}^{2} \mathbf{I}\right], \tag{2} \end{align*}

where as before $\sigma_{n}^{2}$ is the noise variance that accounts for the fact that the observed data are not exactly fitted by the model.

Prior distribution

We define a prior distribution over the parameters. As for the parameter-space perspective, we choose a spherical normal distribution with mean zero, and variance $\sigma_{p}^{2}$:

\begin{equation*} \operatorname{Pr}(\boldsymbol\phi)=\operatorname{Norm}_{\boldsymbol{\phi}}\left[\mathbf{0}, \sigma_{p}^{2} \mathbf{I}\right]. \tag{3} \end{equation*}

This is illustrated in figure 2a

Figure 2. Prior over parameters for 1D regression model. a) The prior for the parameters $\phi_{0}, \phi_{1}$ of a 1D regression model $y=\phi_{0} x+\phi_{1}$ is typically defined as a standard normal distribution with mean zero and a broad covariance $\sigma_{p}^{2} \mathbf{I}$. b) Five samples from the prior; each represents a single choice of parameters $\phi$ and hence defines an input/output mapping $\mathrm{f}[\mathbf{x}, \boldsymbol{\phi}]$. These examples, can alternatively be drawn directly from a prior $\operatorname{Pr}(\mathbf{f})$ over the functions space. c) The model outputs y typically have added noise with variance $\sigma_{w}^{2}$.

Gaussian processes

Now we note that the function values $\mathbf{f}=\mathbf{X}^{T} \boldsymbol{\phi}$ are an affine transform of the parameters, and we use the relation (see proof at end of blog) that if $\operatorname{Pr}(\mathbf{z})=$$\operatorname{Norm}_{\mathbf{z}}[\boldsymbol{\mu}, \boldsymbol{\Sigma}]$, then the affine transform $\mathbf{a}+\mathbf{B z}$ is distributed as:

\begin{equation*} \operatorname{Pr}(\mathbf{a}+\mathbf{B z})=\operatorname{Norm}\left[\mathbf{a}+\mathbf{B} \boldsymbol{\mu}, \mathbf{B} \boldsymbol{\Sigma} \mathbf{B}^{T}\right]. \tag{4} \end{equation*}

Applying this relation, we can get a prior distribution over the function values $\mathbf{f}$:

\begin{equation*} \operatorname{Pr}(\mathbf{f} \mid \mathbf{X})=\operatorname{Norm}_{\mathbf{f}}\left[\mathbf{0}, \sigma_{p}^{2} \mathbf{X}^{T} \mathbf{X}\right]. \tag{5} \end{equation*}

This is technically a Gaussian process, and can equivalently be written as:

\begin{equation*} \mathbf{f}[\mathbf{X}] \sim \mathcal{G P}\left[\mathbf{0}, \sigma_{p}^{2} \mathbf{X}^{T} \mathbf{X}\right]. \tag{6} \end{equation*}

The term Gaussian process just refers to a collection of random variables, any finite number of which have a joint Gaussian distribution (see Rasmussen and Williams, 2006 ). In this case, we can define any input values $\mathbf{X}$ and equation 6 defines a Gaussian distribution between them. For example, we can define a set of equally spaced 1D points and draw samples $\mathbf{f}$ from this prior. Each sample looks like a straight line (figure 2b).

Since the output $\mathbf{y}$ is just a noisy version of the function values, we can similarly write a prior distribution over the output values:

\begin{equation*} \operatorname{Pr}(\mathbf{y} \mid \mathbf{X})=\operatorname{Norm}_{\mathbf{f}}\left[\mathbf{0}, \sigma_{p}^{2} \mathbf{X}^{T} \mathbf{X}+\sigma_{n}^{2} \mathbf{I}\right]. \tag{7} \end{equation*}

Figure 2c illustrates samples drawn from this distribution for a set of equally spaced inputs.

Consider a test set of points $\mathbf{X}^{*}$ which map to function values $\mathbf{f}^{*}$. We can write the joint distribution of the training values $\mathbf{y}$ and the new function value $\mathbf{f}^{*}$:

\[ \operatorname{Pr}\binom{\mathbf{y}}{\mathbf{f}^{*}}=\operatorname{Norm}_{\left[\mathbf{y},\mathbf{f}^{*} \right]^{T}}\left[\left[\begin{array}{l} 0 \tag{8}\\ 0 \end{array}\right],\left[\begin{array}{cc} \sigma_{p}^{2} \mathbf{X}^{* T} \mathbf{X}^{*} & \sigma_{p}^{2} \mathbf{X}^{* T} \mathbf{X} \\ \sigma_{p}^{2} \mathbf{X}^{T} \mathbf{X}^{*} & \sigma_{p}^{2} \mathbf{X}^{T} \mathbf{X}+\sigma_{n}^{2} \mathbf{I} \end{array}\right]\right]. \]

Now we use the following well-known relation for conditional distributions in Gaussian variables (proof at end of blog): if two sets of variables $\mathbf{z}_{1}$ and $\mathbf{z}_{2}$ are jointly distributed as a multivariate normal:

\[ \operatorname{Pr}\left(\left[\begin{array}{l} \mathbf{z}_{1} \tag{9}\\ \mathbf{z}_{2} \end{array}\right]\right)=\operatorname{Norm}_{\mathbf{z}}\left[\left[\begin{array}{l} \boldsymbol{\mu}_{1} \\ \boldsymbol{\mu}_{2} \end{array}\right],\left[\begin{array}{ll} \boldsymbol{\Sigma}_{11} & \boldsymbol{\Sigma}_{12} \\ \boldsymbol{\Sigma}_{21} & \boldsymbol{\Sigma}_{22} \end{array}\right]\right], \]

then the conditional distribution of $\mathbf{z}_{2}$ given observed values $\mathbf{z}_{1}$ is:

\begin{equation*} \operatorname{Pr}\left(\mathbf{z}_{1} \mid \mathbf{z}_{2}\right)=\operatorname{Norm}_{\mathbf{z}_{1}}\left[\boldsymbol{\mu}_{\mathbf{1}}+\boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1}\left(\mathbf{z}_{2}-\boldsymbol{\mu}_{2}\right), \boldsymbol{\Sigma}_{11}-\boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12}^{T}\right]. \tag{10} \end{equation*}

Applying this relation to equation 8, we get:

\begin{align*} \operatorname{Pr}\left(\mathbf{f}^{*} \mid \mathbf{X}^{*}, \mathbf{X}, \mathbf{y}\right)&= \operatorname{Norm}_{\mathbf{f}^{*}}\left[\sigma_{p}^{2} \mathbf{X}^{* T} \mathbf{X}\left(\sigma_{p}^{2} \mathbf{X}^{T} \mathbf{X}+\sigma_{n}^{2} \mathbf{I}\right)^{-1} \mathbf{y}\right., \tag{11}\\ &\hspace{3cm}\left.\sigma_{p}^{2} \mathbf{X}^{* T} \mathbf{X}^{*}-\sigma_{p}^{2} \mathbf{X}^{* T} \mathbf{X}\left(\sigma_{p}^{2} \mathbf{X}^{T} \mathbf{X}+\sigma_{n}^{2} \mathbf{I}\right)^{-1} \sigma_{p}^{2} \mathbf{X}^{T} \mathbf{X}^{*}\right] \\ &= \operatorname{Norm}_{\mathbf{f}^{*}}\Biggl[\mathbf{X}^{* T} \mathbf{X}\left(\mathbf{X}^{T} \mathbf{X}+\frac{\sigma_{n}^{2}}{\sigma_{p}^{2}} \mathbf{I}\right)^{-1} \mathbf{y}, \\ &\hspace{3cm}\sigma_{p}^{2} \mathbf{X}^{* T} \mathbf{X}^{*}-\sigma_{p}^{2} \mathbf{X}^{* T} \mathbf{X}\left(\mathbf{X}^{T} \mathbf{X}+\frac{\sigma_{n}^{2}}{\sigma_{p}^{2}} \mathbf{I}\right)^{-1} \mathbf{X}^{T} \mathbf{X}^{*}\Biggr] \end{align*}

where we have moved the terms $\sigma_{p}^{2}$ into the inverse in the second line. If we draw samples from this distribution for equally spaced test points in $\mathbf{X}^{*}$, each sample looks like a line that is broadly compatible with the training data $\mathbf{X}$ (figure 3a).

Figure 3. Samples from posterior distribution over functions $\operatorname{Pr}\left(\mathbf{f}^{*} \mid \mathbf{X}^{*}, \mathbf{X}, \mathbf{y}\right)$ conditioned on six observed data points $\left\{\mathbf{x}_{i}, y_{i}\right\}$ (stored in $\mathbf{X}$ and $\mathbf{y}$. Each looks like a line that is broadly compatible with the data. b) Samples from output function; these are the same, but with added noise $\sigma_{n}^{2}$; we expect the observed data points to be samples from this distribution.

Finally, we note that the original data points are drawn from distribution over the noisy output $\mathbf{y}^{*}$, which has the same distribution plus an the extra noise term on the covariance.

\begin{align*} & \operatorname{Pr}\left(\mathbf{y}^{*} \mid \mathbf{X}^{*}, \mathbf{X}, \mathbf{y}\right)=\operatorname{Norm}_{\mathbf{y}^{*}} {\Biggl[\mathbf{X}^{* T} \mathbf{X}\left(\mathbf{X}^{T} \mathbf{X}+\frac{\sigma_{n}^{2}}{\sigma_{p}^{2}} \mathbf{I}\right)^{-1} \mathbf{y}} \\ &\hspace{3cm}\sigma_{p}^{2} \mathbf{X}^{* T} \mathbf{X}^{*}-\sigma_{p}^{2} \mathbf{X}^{* T} \mathbf{X}\left(\mathbf{X}^{T} \mathbf{X}+\frac{\sigma_{n}^{2}}{\sigma_{p}^{2}} \mathbf{I}\right)^{-1} \mathbf{X}^{T} \mathbf{X}^{*}+\sigma_{n}^{2} \mathbf{I}\Biggl]. \tag{12} \end{align*}

Samples from this distribution look like noisy lines that are broadly compatible with the data points (figure 3b).

Figure 4 visualizes the solution in a different way. Here, we have plotted the mean of the posterior (the MAP solution) and the marginal uncertainty at each of a set of equally spaced positions (represented by a $\pm 2$ standard deviation region). In this way, we can visualize the function and its uncertainty, or the equivalent result in output space.

Figure 4. Posterior distribution with marginal uncertainty. a) The function predictions can also be visualized as the mean of the posterior distribution (the MAP solution, cyan line) and the marginal uncertainty at each input point (represented by $\pm 2$ standard deviation region). The uncertainty increases as we extrapolate further from the data. b) The noisy output predictions can be visualized in the same way but have additional noise $\sigma_{n}^{2}$ at each point. The observed data points will usually fall within the $\pm 2$ standard deviation region.

Parameter space vs. function space

Recall from part IV of this series of blogs that the predictive distribution for the parameter space (Bayesian) view of the linear model was:

\begin{align*} \operatorname{Pr}\left(y^{*} \mid \mathbf{x}^{*}, \mathbf{X}, \mathbf{y}\right)=\operatorname{Norm}_{y^{*}}\Biggl[ & \mathbf{x}^{* T} \mathbf{X}\left(\mathbf{X}^{T} \mathbf{X}+\frac{\sigma_{n}^{2}}{\sigma_{p}^{2}} \mathbf{I}\right)^{-1} \mathbf{y} \tag{13}\\ & \sigma_{p}^{2} \mathbf{x}^{* T} \mathbf{x}^{*}-\sigma_{p}^{2} \mathbf{x}^{* T} \mathbf{X}\left(\mathbf{X}^{T} \mathbf{X}+\frac{\sigma^{2}}{\sigma_{p}^{2}} \mathbf{I}\right)^{-1} \mathbf{X}^{T} \mathbf{x}^{*}+\sigma_{n}^{2}\Biggr]. \end{align*}

It’s clear that for this model, the parameter space view and the function space view lead to the same result (although we developed the function space approach to make multiple predictions simultaneously). However, this is NOT true in general. In future articles in this series, we will see that these two perspectives have quite different implications for neural networks.

Non-linear regression

For both the parameter space and function space views, we only yield neat closed-form solution for the predictions because the model is linear; although the same principles apply for non-linear models, there is not generally a closed-form solution. What then, should we do when faced with a training dataset for which a linear model is clearly inadequate? One tractable approach is to transform the initial data by a fixed non-linear transformation $\mathbf{z}=\mathbf{g}[\mathbf{x}]$ and then apply a linear model $\mathrm{f}[\mathbf{z}, \boldsymbol{\phi}]$ to the transformed data.

Example: polynomial regression

A simple example of this type of model is polynomial regression for 1D data $x$ in which we might use the non-linear transformation (figure 2a):

\[ \mathbf{z}=\left[\begin{array}{c} 1 \tag{2}\\ x \\ x^{2} \\ x^{3} \\ x^{4} \end{array}\right]. \]

If we draw samples from the prior from this model, each looks like a polynomial of order five (figure 5).

Figure 5. Polynomial model. a) The input data $x$ is transformed by a fixed non-linearity to create a new input vector $\mathbf{z}$ which is then passed into the linear model. Here the 1D data $x$ is transformed to a 5D vector consisting of the first four polynomial basis functions and the constant function (not shown). b) If we define a prior over the model parameters $\phi$, then we can draw samples from the induced prior over functions. Each looks like a random fourth-order polynomial.

When we condition the output function on a training dataset, we samples from the posterior look like order-5 polynomial curves that broadly agree with the data (figure 6a), or we can alternatively visualize the MAP solution and the associated marginal uncertainty (figure 6b).

Figure 6. Posterior distribution for polynomial model. a) If we condition the function on a set of observed data points, we can compute a posterior distribution over the function values. Each curve represents a sample from this distribution. b) As before, we can visualize this distribution using the MAP sample (blue line) and the marginal variance (gray region).

In this way, we can make closed-form predictions based on a polynomial model and yield estimates of uncertainty, without ever explicitly ‘fitting’ the model and committing to a single set of parameters.

Example: ReLU model

A second example is a transformation based on a collection of offset ReLU functions :

\[ \mathbf{z}=\left[\begin{array}{c} 1 \tag{15}\\ \operatorname{ReLU}\left[\beta_{1}+\omega_{1} x\right] \\ \operatorname{ReLU}\left[\beta_{2}+\omega_{2} x\right] \\ \operatorname{ReLU}\left[\beta_{3}+\omega_{3} x\right] \\ \operatorname{ReLU}\left[\beta_{4}+\omega_{4} x\right] \\ \operatorname{ReLU}\left[\beta_{5}+\omega_{5} x\right] \end{array}\right]. \]

Note that this is not exactly the same as a neural network because in this case the biases $\beta_{0}$ and slopes $\omega_{0}$ are chosen in advance and fixed; they are not optimized (or marginalized over) as part of the training process.

The prior and posterior functions are visualized in figure 7 and 8, respectively. In each case the output function is a piecewise linear, with the position of the joints determined by the joints in the component ReLU functions (which are themselves determined by the predefined $\beta_{\bullet}$ and $\omega_{\bullet}$ parameters.

Figure 7. ReLU model. a) The input data $x$ is transformed by a fixed non-linearity to create a new input vector $\mathbf{z}$ which is then passed into the linear model. Here the 1D data $x$ is transformed to a 6D vector consisting of the five horizontally offset ReLU functions and the constant function (not shown). b) If we define a prior over the model parameters $\phi$, then we can draw samples from the induced prior over functions. Each looks like a piecewise linear function where the position of the joints is determined by the component ReLU functions.

Figure 8. Posterior distribution for ReLU model. a) If we condition the function on a set of observed data points, we can compute a posterior distribution over the function values. Each curve represents a sample from this distribution. b) As before, we can visualize this distribution using the MAP sample (blue line) and the marginal variance (gray region).

Disadvantages of non-linear regression

Although, the approach of transforming the input data using a fixed non-linearity is simple and effective, these methods do not necessarily scale well to high dimensions. For example, if we have 1000 dimensional input, then the set of third degree polynomials contains $\left(D_{z}=1+1000+1000^{2}+1000^{3}\right)$ terms. However, the complexity of the fitted function should ultimately depend on the amount of training data. If this is significantly less than the number of transformed dimensions, we should be able to make efficiency gains.

Kernels and the kernel trick

To make progress, we note that the final equation for the predictive distribution:

\begin{align*} \operatorname{Pr}\left(\mathbf{y}^{*} \mid \mathbf{Z}^{*}, \mathbf{Z}, \mathbf{y}\right)=\operatorname{Norm}_{\mathbf{y}^{*}} & {\Biggl[\mathbf{Z}^{* T} \mathbf{Z}\left(\mathbf{Z}^{T} \mathbf{Z}+\frac{\sigma_{n}^{2}}{\sigma_{p}^{2}} \mathbf{I}\right)^{-1} \mathbf{y}}, \\ & \sigma_{p}^{2} \mathbf{Z}^{* T} \mathbf{Z}^{*}-\sigma_{p}^{2} \mathbf{Z}^{* T} \mathbf{Z}\left(\mathbf{Z}^{T} \mathbf{Z}+\frac{\sigma_{n}^{2}}{\sigma_{p}^{2}} \mathbf{I}\right)^{-1} \mathbf{Z}^{T} \mathbf{Z}^{*}+\sigma_{n}^{2} \mathbf{I}\Biggr] \tag{16} \end{align*}

depends only on matrix products of the form $\mathbf{Z}^{T} \mathbf{Z}$ where $\mathbf{Z} \in \mathbb{R}^{D_{z} \times I}$ contains the transformed data vectors in its columns.

When the dimension $D_{z}$ of the transformed vectors $\mathbf{z}=\mathbf{g}[\mathbf{x}]$ is high, computing the non-linear transformations and computing the dot products that make the matrix product can be time-consuming. A viable alternative is kernel substitution in which we directly define a single kernel function $\mathrm{k}\left[\mathbf{x}_{i}, \mathbf{x}_{j}\right]$ as a replacement for the operations $\mathbf{g}\left[\mathbf{x}_{i}\right]^{T} \mathbf{g}\left[\mathbf{x}_{j}\right]$. For many transformations $\mathbf{g}[\bullet]$ it is more efficient to evaluate the kernel function directly than to transform the variables separately and then compute the dot product.

Taking this idea one step further, it is possible to choose a kernel function $\mathrm{k}\left[\mathbf{x}_{i}, \mathbf{x}_{j}\right]$ without knowledge of what transformation $\mathbf{g}[\bullet]$ it corresponds to. Clearly, the function must be carefully chosen so that it does in fact correspond to computing some function $\mathbf{z}=\mathbf{g}[\mathbf{x}]$ of each data vector and taking the inner product of the results: for example, since $\mathbf{z}_{i}^{T} \mathbf{z}_{j}=\mathbf{z}_{j}^{T} \mathbf{z}_{i}$ it must treat its arguments symmetrically so that $\mathrm{k}\left[\mathbf{x}_{i}, \mathbf{x}_{j}\right]=\mathrm{k}\left[\mathbf{x}_{j}, \mathbf{x}_{i}\right]$.

When we use kernel functions, we no longer explicitly compute the transformed vector $\mathbf{z}$. One advantage of this is we can define kernel functions that correspond to projecting the data into very high dimensional or even infinite spaces.

We now return to the question of which constraints must be put on the kernel function to ensure that there exists an underlying function $\mathbf{f}\left[\mathbf{x}_{i}\right]=\mathbf{z}_{i}$ so that $\mathrm{k}\left[\mathbf{x}_{i}, \mathbf{x}_{j}\right]=\mathbf{z}_{i}^{T} \mathbf{z}_{j}$. Mercer’s theorem states that this is true when the kernel’s arguments are in a measurable space, and the kernel is positive semi-definite so that:

\begin{equation*} \sum_{i j} \mathrm{k}\left[\mathbf{x}_{i}, \mathbf{x}_{j}\right] a_{i} a_{j} \geq 0, \tag{17} \end{equation*}

for any finite subset $\left\{\mathbf{x}_{n}\right\}_{n=1}^{N}$ of vectors in the space and any real numbers $\left\{a_{n}\right\}_{n=1}^{N}$. Examples of valid kernels include:

• Linear: \begin{equation*} \mathrm{k}\left[\mathbf{x}_{i}, \mathbf{x}_{j}\right]=\mathbf{x}_{i}^{T} \mathbf{x} \tag{18} \end{equation*}
• Degree $p$ polynomial : \begin{equation*} \mathrm{k}\left[\mathbf{x}_{i}, \mathbf{x}_{j}\right]=\left(\mathbf{x}_{i}^{T} \mathbf{x}_{j}+1\right)^{p} \tag{19} \end{equation*}
• Radial basis function (RBF) or Gaussian: \begin{equation*} \mathrm{k}\left[\mathbf{x}_{i}, \mathbf{x}_{j}\right]=\exp \left[-0.5\left(\frac{\left(\mathbf{x}_{i}-\mathbf{x}_{j}\right)^{T}\left(\mathbf{x}_{i}-\mathbf{x}_{j}\right)}{\lambda^{2}}\right)\right] \tag{20} \end{equation*}

The last of these is particularly interesting. It can be shown that this kernel function corresponds to computing infinite length vectors $\mathbf{z}$ and taking their dot product. The entries of $\mathbf{z}$ correspond to evaluating a radial basis function at every possible point in the space of $\mathbf{x}$.

It is also possible to create new kernels by combining two or more existing kernels. For example, sums and products of valid kernels are guaranteed to be positive semi-definite and so are also valid kernels.

Using kernel notation, we can now write the predictive equation as:

\begin{align*} & \operatorname{Pr}\left(\mathbf{y}^{*} \mid \mathbf{X}^{*}, \mathbf{X}, \mathbf{y}\right)=\operatorname{Norm}_{\mathbf{y}^{*}}\Biggl[\mathbf{K}[\mathbf{X}, \mathbf{X}]\left(\mathbf{K}[\mathbf{X}, \mathbf{X}]+\frac{\sigma_{n}^{2}}{\sigma_{p}^{2}} \mathbf{I}\right)^{-1} \mathbf{y}, \\ & \hspace{2cm}\quad \sigma_{p}^{2} \mathbf{K}\left[\mathbf{X}^{*}, \mathbf{X}^{*}\right]-\sigma_{p}^{2} \mathbf{K}\left[\mathbf{X}^{*}, \mathbf{X}\right]\left(\mathbf{K}[\mathbf{X}, \mathbf{X}]+\frac{\sigma_{n}^{2}}{\sigma_{p}^{2}} \mathbf{I}\right)^{-1} \mathbf{K}\left[\mathbf{X}, \mathbf{X}^{*}\right]+\sigma_{n}^{2} \mathbf{I}\Biggr], \tag{21} \end{align*}

where the $(i, j)^{\text {th }}$ element of $\mathbf{K}\left[\mathbf{X}^{*}, \mathbf{X}\right]$ would comprise the element $\mathrm{k}\left[\mathbf{x}_{i}^{*}, \mathbf{x}_{j}\right]$ and so on. Figure 2.9 shows the prior and posterior distribution for kernel regression using the RBF kernel; it results in a set of smooth functions that pass close to the data points.

Figure 9. Kernel regression with RBF kernel. a) Samples from the prior distribution over functions look like smooth curves (the speed of change is determined by the kernel parameter $\lambda$). b) Samples from the posterior look like smooth functions that pass through the training data. c) This posterior can also be visualized as the MAP function and the marginal uncertainty.

In this blog we described a method to define a prior over the space of output functions. We can use this prior find a joint distribution over the training and test data points; we then find the conditional distribution over the test points, given the training points. For linear models, this gives the same results as the parameter-space perspective on Bayesian modelling. We saw how this approach can be extended to model more complex data by applying a non-linear transformation to the input data before applying the linear model.

The next part of this series of blogs applies the Bayesian approach to deep neural networks. We will see that the implications of parameter-space and function-space perspectives are quite different here.

Affine transformations of normal distributions

The multivariate normal distribution is defined as:

$$ \operatorname{Pr}(\mathbf{x})=\frac{1}{(2 \pi)^{D}|\boldsymbol{\Sigma}|^{1 / 2}} \exp \left[-0.5(\mathbf{x}-\boldsymbol{\mu}) \boldsymbol{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})\right]. $$

To transform the distribution as $\mathbf{y}=\mathbf{A} \mathbf{x}+\mathbf{b}$, we substitute in $\mathbf{x}=\mathbf{A}^{-1}(\mathbf{y}-\mathbf{b})$ and divide by the determinant of the Jacobian of the transformation, which for this case is just $|\mathbf{A}|$ giving:

$$ \operatorname{Pr}(\mathbf{x})=\frac{1}{(2 \pi)^{D}|\boldsymbol{\Sigma}|^{1 / 2}|\mathbf{A}|} \exp \left[-0.5\left(\mathbf{A}^{-1}(\mathbf{y}-\mathbf{b})-\boldsymbol{\mu}\right) \boldsymbol{\Sigma}^{-1}\left(\mathbf{A}^{-1}(\mathbf{y}-\mathbf{b})-\boldsymbol{\mu}\right)\right]. $$

We’ll first work with the constant $\kappa$ from outside the exponential to show that it is the constant for a distribution with variance $\mathbf{A} \boldsymbol{\Sigma} \mathbf{A}^{T}$. We have:

\begin{aligned} \kappa & =\frac{1}{(2 \pi)^{D}|\mathbf{\Sigma}|^{1 / 2}|\mathbf{A}|} \\ & =\frac{1}{(2 \pi)^{D}|\mathbf{\Sigma}|^{1 / 2}|\mathbf{A}|^{1 / 2}|\mathbf{A}|^{1 / 2}} \\ & =\frac{1}{(2 \pi)^{D}|\mathbf{\Sigma}|^{1 / 2}|\mathbf{A}|^{1 / 2}\left|\mathbf{A}^{T}\right|^{1 / 2}} \\ & =\frac{1}{(2 \pi)^{D}|\mathbf{A}|^{1 / 2}|\mathbf{\Sigma}|^{1 / 2}\left|\mathbf{A}^{T}\right|^{1 / 2}} \\ & =\frac{1}{(2 \pi)^{D}\left|\mathbf{A} \mathbf{\Sigma} \mathbf{A}^{T}\right|^{1 / 2}}, \end{aligned}

which is exactly the required constant.

Now we’ll work with the quadratic in the exponential term to show that it corresponds to a normal distribution in $\mathbf{y}$ with variance $\mathbf{A} \boldsymbol{\Sigma} \mathbf{A}^{T}$ and mean $\mathbf{A} \boldsymbol{\mu}+\mathbf{b}$.

\begin{aligned} \left(\mathbf{A}^{-1}(\mathbf{y}-\mathbf{b})-\boldsymbol{\mu}\right) \boldsymbol{\Sigma}^{-1}\left(\mathbf{A}^{-1}(\mathbf{y}-\mathbf{b})-\boldsymbol{\mu}\right) &\\ &\hspace{-4cm}=\mathbf{y}^{T} \mathbf{A}^{-T} \boldsymbol{\Sigma}^{-1} \mathbf{A}^{-1} \mathbf{y}-2\left(\mathbf{b}^{T} \mathbf{A}^{-T}+\boldsymbol{\mu}^{T}\right) \boldsymbol{\Sigma}^{-1} \mathbf{A}^{-1} \mathbf{y} \\ &+\left(\mathbf{A}^{-1} \mathbf{b}+\boldsymbol{\mu}\right)^{T} \boldsymbol{\Sigma}^{-1}\left(\mathbf{A}^{-1} \mathbf{b}+\boldsymbol{\mu}\right) \\ &\hspace{-4cm}=\mathbf{y}^{T} \mathbf{A}^{-T} \boldsymbol{\Sigma}^{-1} \mathbf{A}^{-1} \mathbf{y}-2\left(\mathbf{b}^{T}+\boldsymbol{\mu}^{T} \mathbf{A}^{T}\right) \mathbf{A}^{-T} \boldsymbol{\Sigma}^{-1} \mathbf{A}^{-1} \mathbf{y} \\ &+\left(\mathbf{A}^{-1} \mathbf{b}+\mathbf{A} \boldsymbol{\mu}\right)^{T} \mathbf{A}^{-T} \boldsymbol{\Sigma}^{-1} \mathbf{A}^{-1}(\mathbf{b}+\mathbf{A} \boldsymbol{\mu}) \\ &\hspace{-4cm}=(\mathbf{y}-(\mathbf{A} \boldsymbol{\mu}+\mathbf{b}))^{T}\left(\mathbf{A} \boldsymbol{\Sigma} \mathbf{A}^{T}\right)^{-1}(\mathbf{y}-(\mathbf{A} \boldsymbol{\mu}+\mathbf{b})). \end{aligned}

This is clearly the quadratic term from a normal distribution in $\mathbf{y}$ with variance $\mathbf{A} \boldsymbol{\Sigma} \mathbf{A}^{T}$ and mean $\mathbf{A} \boldsymbol{\mu}+\mathbf{b}$ as required.

Conditional distribution of a Gaussian

Consider a joint Gaussian distribution with variables $\mathbf{z}=\left[\mathbf{z}_{1}, \mathbf{z}_{2}\right]$:

$$\operatorname{Pr}(\mathbf{z})=\operatorname{Pr}\left(\left[\begin{array}{l} \mathbf{z}_{1} \\ \mathbf{z}_{2} \end{array}\right]\right)=\operatorname{Norm}_{\mathbf{x}}\left[\left[\begin{array}{l} \boldsymbol{\mu}_{1} \\ \boldsymbol{\mu}_{2} \end{array}\right],\left[\begin{array}{ll} \boldsymbol{\Sigma}_{11} & \boldsymbol{\Sigma}_{12} \\ \boldsymbol{\Sigma}_{21} & \boldsymbol{\Sigma}_{22} \end{array}\right]\right].$$

The conditional distribution of the varaibles $\mathbf{z}_{1}$ given observed variables $\mathbf{z}_{2}$ is:

\begin{equation*} \operatorname{Pr}\left(\mathbf{z}_{1} \mid \mathbf{z}_{2}\right)=\operatorname{Norm}_{\mathbf{z}_{1}}\left[\boldsymbol{\mu}_{\mathbf{1}}+\boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1}\left(\mathbf{z}_{2}-\boldsymbol{\mu}_{2}\right), \boldsymbol{\Sigma}_{11}-\boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12}^{T}\right] \end{equation*}

Proof: We can write out the joint distribution as:

\begin{aligned} \operatorname{Pr}\left(\left[\begin{array}{l} \mathbf{z}_{1} \\ \mathbf{z}_{2} \end{array}\right]\right) & =\kappa_{1} \exp \left[-0.5\left(\left[\begin{array}{l} \mathbf{z}_{1} \\ \mathbf{z}_{2} \end{array}\right]-\left[\begin{array}{l} \boldsymbol{\mu}_{1} \\ \boldsymbol{\mu}_{2} \end{array}\right]\right)^{T}\left[\begin{array}{ll} \boldsymbol{\Sigma}_{11} & \boldsymbol{\Sigma}_{12} \\ \boldsymbol{\Sigma}_{12}^{T} & \mathbf{\Sigma}_{22} \end{array}\right]^{-1}\left(\left[\begin{array}{l} \mathbf{z}_{1} \\ \mathbf{z}_{2} \end{array}\right]-\left[\begin{array}{l} \boldsymbol{\mu}_{1} \\ \boldsymbol{\mu}_{2} \end{array}\right]\right)\right] \\ & =\kappa_{1} \exp \left[-0.5\left(\left[\begin{array}{l} \mathbf{z}_{1} \\ \mathbf{z}_{2} \end{array}\right]-\left[\begin{array}{l} \boldsymbol{\mu}_{1} \\ \boldsymbol{\mu}_{2} \end{array}\right]\right)^{T}\left[\begin{array}{ll} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{array}\right]\left(\left[\begin{array}{l} \mathbf{z}_{1} \\ \mathbf{z}_{2} \end{array}\right]-\left[\begin{array}{l} \boldsymbol{\mu}_{1} \\ \boldsymbol{\mu}_{2} \end{array}\right]\right)\right], \end{aligned}

where $\kappa_{1}$ is the standard constant associated with the normal distribution and the terms $\mathbf{A}, \mathbf{B}, \mathbf{C}$ and $\mathbf{D}$ are the four components of the inverse covariance matrix. We can find these using Schur’s complement relation to write:

\begin{aligned} & \mathbf{A}=\left(\boldsymbol{\Sigma}_{11}-\boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12}^{T}\right)^{-1} \\ & \mathbf{B}=-\left(\boldsymbol{\Sigma}_{11}-\boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12}^{T}\right)^{-1} \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} \\ & \mathbf{C}=-\boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12}^{T}\left(\boldsymbol{\Sigma}_{11}-\boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12}^{T}\right)^{-1} \\ & \mathbf{D}=\boldsymbol{\Sigma}_{22}^{-1}+\boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12}^{T}\left(\boldsymbol{\Sigma}_{11}-\boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12}^{T}\right)^{-1} \boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1}. \end{aligned}

This can easily be confirmed by multiplying the resulting inverse matrix with the original covariance matrix, which yields the identity.

Now we exploit the conditional probability relation $\operatorname{Pr}\left(\mathbf{z}_{1} \mid \mathbf{z}_{2}\right) \operatorname{Pr}\left(\mathbf{z}_{2}\right)=\operatorname{Pr}\left(\mathbf{z}_{1}, \mathbf{z}_{2}\right)$ and note that $\operatorname{Pr}\left(\mathbf{z}_{1} \mid \mathbf{z}_{2}\right) \propto \operatorname{Pr}\left(\mathbf{z}_{1}, \mathbf{z}_{2}\right)$. We hence take the approach of reformulating the joint distribution in terms of a normal distribution in $\mathbf{z}_{1}$ alone. Multiplying out the matrices in the exponent, we see that:

$$\operatorname{Pr}\left(\mathbf{z}_{1} \mid \mathbf{z}_{2}\right)=\kappa_{2} \cdot \exp \left[-0.5\left(\left(\mathbf{z}_{1}-\boldsymbol{\mu}_{1}\right)^{T} \mathbf{A}\left(\mathbf{z}_{1}-\boldsymbol{\mu}_{1}\right)+2\left(\mathbf{z}_{1}-\boldsymbol{\mu}_{1}\right)^{T} \mathbf{B}\left(\mathbf{z}_{2}-\boldsymbol{\mu}_{2}\right)\right)\right],$$

where we have subsumed the terms that do not depend on $\mathbf{z}_{1}$ to create a new constant $\kappa_{2}$. Now we both multiply and divide by a new term to complete the square:

\begin{array}{r} \operatorname{Pr}\left(\mathbf{z}_{1} \mid \mathbf{z}_{2}\right)=\kappa_{3} \cdot \exp \left[-0.5\left(\left(\mathbf{z}_{1}-\boldsymbol{\mu}_{1}\right)^{T} \mathbf{A}\left(\mathbf{z}_{1}-\boldsymbol{\mu}_{1}\right)+2\left(\mathbf{z}_{1}-\boldsymbol{\mu}_{1}\right)^{T} \mathbf{B}\left(\mathbf{z}_{2}-\boldsymbol{\mu}_{2}\right)\right.\right. \\ \left.\left.+\left(\mathbf{z}_{2}-\boldsymbol{\mu}_{2}\right)^{T} \mathbf{B}^{T} \mathbf{A}^{-1} \mathbf{B}\left(\mathbf{z}_{2}-\boldsymbol{\mu}_{2}\right)\right)\right], \end{array}

where the constant $\kappa_{3}$ has changed by a factor that balances the extra term in the exponential. Writing this out as a normal distribution, we have:

\begin{equation*} \operatorname{Pr}\left(\mathbf{z}_{1} \mid \mathbf{z}_{2}\right)=\kappa_{3} \cdot \operatorname{Norm}_{\mathbf{z}_{1}}\left[\boldsymbol{\mu}_{1}+\mathbf{A}^{-1} \mathbf{B}\left(\mathbf{z}_{2}-\boldsymbol{\mu}_{2}\right), \mathbf{A}^{-1}\right]. \end{equation*}

The constant of proportionality must be $1 /\left(\kappa_{3}|\mathbf{D}|^{1 / 2} 2 \pi^{D / 2}\right)$ so that it is a proper normal distribution. Hence the conditional distribution is given by:

\begin{align*} \operatorname{Pr}\left(\mathbf{z}_{1} \mid \mathbf{z}_{2}\right) & =\operatorname{Norm}_{\mathbf{z}_{1}}\left[\boldsymbol{\mu}_{1}-\mathbf{A}^{-1} \mathbf{B}\left(\mathbf{z}_{2}-\boldsymbol{\mu}_{2}\right), \mathbf{A}^{-1}\right]\\ & =\operatorname{Norm}_{\mathbf{z}_{2}}\left[\boldsymbol{\mu}_{1}+\boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1}\left(\mathbf{z}_{2}-\boldsymbol{\mu}_{2}\right), \boldsymbol{\Sigma}_{11}-\boldsymbol{\Sigma}_{12} \boldsymbol{\Sigma}_{22}^{-1} \boldsymbol{\Sigma}_{12}^{T}\right], \end{align*}

as required.

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