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[Users](https://mathoverflow.net/users) # [Bayesian statistics for pure mathematicians](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians) [Ask Question](https://mathoverflow.net/questions/ask) Asked 13 years, 7 months ago Modified [2 years, 9 months ago](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians?lastactivity "2023-04-25 06:41:13Z") Viewed 8k times This question shows research effort; it is useful and clear 32 Save this question. Show activity on this post. Could someone please recommend reading on Bayesian statistics presented from a pure mathematical point of view? That is, works that start assuming a good knowledge of measure theoretic probability. The usual reference works for statisticians gloss over the fine details, but for me this just leads to more confusion\! I already know of the following question [Statistics for mathematicians](https://mathoverflow.net/questions/31655/statistics-for-mathematicians) and will follow up some of the suggestions there, but I am specifically looking for a treatment of the Bayesian approach, so if there are any recommendations that apply just to this I would appreciate hearing about them. - [pr.probability](https://mathoverflow.net/questions/tagged/pr.probability "show questions tagged 'pr.probability'") - [st.statistics](https://mathoverflow.net/questions/tagged/st.statistics "show questions tagged 'st.statistics'") - [reference-request](https://mathoverflow.net/questions/tagged/reference-request "show questions tagged 'reference-request'") [Share](https://mathoverflow.net/q/100288 "Short permalink to this question") Share a link to this question Copy link [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/ "The current license for this post: CC BY-SA 3.0") Cite [Improve this question](https://mathoverflow.net/posts/100288/edit) Follow Follow this question to receive notifications [edited Apr 13, 2017 at 12:58](https://mathoverflow.net/posts/100288/revisions "show all edits to this post") community wiki [Tom Ellis](https://mathoverflow.net/posts/100288/revisions "show revision history for this post") 4 - 1 I would say that engineering books are more easy to read and to get main ideas, examples and motivations. May be like this one [amazon.com/…](https://rads.stackoverflow.com/amzn/click/com/0133457117) Alexander Chervov – [Alexander Chervov](https://mathoverflow.net/users/10446/alexander-chervov) 2012-06-22 15:12:13 +00:00 [Commented Jun 22, 2012 at 15:12](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians#comment257950_100288) - Tom, did you end up finding a book to your liking? I too wish to understand the Bayesian framework from a rigorous, pure math standpoint, so I'd appreciate any suggestions or advice. goblin GONE – [goblin GONE](https://mathoverflow.net/users/26080/goblin-gone) 2016-05-02 09:11:55 +00:00 [Commented May 2, 2016 at 9:11](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians#comment588442_100288) - Hi @goblin, I'm afraid I can't recommend anything specific. Tom Ellis – [Tom Ellis](https://mathoverflow.net/users/3676/tom-ellis) 2016-05-05 10:21:15 +00:00 [Commented May 5, 2016 at 10:21](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians#comment589048_100288) - 1 Two recommendations: [*The Frequentist Theory of Bayesian Statistics* by Kleijn](https://staff.fnwi.uva.nl/b.j.k.kleijn/bkleijn-book-work-in-progress-Sep2022.pdf) and [*Bayesian Field Theory* by Lemm](https://arxiv.org/abs/physics/9912005). Durden – [Durden](https://mathoverflow.net/users/141372/durden) 2024-04-22 05:11:58 +00:00 [Commented Apr 22, 2024 at 5:11](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians#comment1219441_100288) [Add a comment](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians "Use comments to ask for more information or suggest improvements. Avoid answering questions in comments.") \| ## 4 Answers 4 Sorted by: [Reset to default](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians?answertab=scoredesc#tab-top) This answer is useful 17 Save this answer. Show activity on this post. Many hold that Bayesian statistics "from a purely mathematical point of view" is entirely coextensive with probability (however it is that you want to define its boundaries as a mathematical discipline). Nonetheless, if I interpret your request as being for a mathematically sophisticated and rigorous exposition on why the Bayesian approach is a worthy one, three book spring to mind. 1. Theory of Statistics by Mark Schervish 2. Bayes Theory by John Hartigan 3. The Bayesian Choice by Christian Robert The first of these is a general graduate text in statistics, but the author gives uncommonly complete coverage of both Bayesian and frequentist methods. The second is a smaller volume and, as I recall, is devoted to some of the more delicate issues surround finite versus countable additivity as relates to using probability distributions as priors in a Bayesian approach. The final book is more general, but the style is more formal than the Bernardo and Smith book mentioned by PaPiro. (This is, in my experience, true of the style of French Bayesians :) As I said, the distinctive elements of the Bayesian perspective are more philosophical than technical, but there are some technical areas that have received attention in the Bayesian community that may be of independent mathematical interest. One would be the role of so-called "improper" priors as mentioned above. Another is the role of conditional distributions as a primitive rather than derived notion, leading to the idea of disintegration, as in this manuscript of [Pollard](http://www.stat.yale.edu/~pollard/Papers/chang-pollard.pdf). Also, because of a keen interest in the application of Monte Carlo methods, Bayesian statisticians have to a lot of work on various aspects of computational methods for sampling from various distributions. Christian Robert is a prominent researcher in this area, and he has a [blog](https://xianblog.wordpress.com/). The current post happens to be about Bayesian foundations. Finally, at the heart of a many arguments in favor of a Bayesian approach (early chapters in Bernardo and Smith and Robert are dedicated to it) are de Finetti type representation theorems, which sanction prior distributions via appeals to exchangeability. You can start with the wiki entry for [de Finetti theorems](https://en.wikipedia.org/wiki/De_Finetti%27s_theorem) and then look at the [work of Persi Diaconis](https://projecteuclid.org/journals/annals-of-probability/volume-8/issue-4/Finite-Exchangeable-Sequences/10.1214/aop/1176994663.full) on the topic. In this vein see also [Lauritzen's monograph](https://books.google.com/books?id=8Pzs_EujC28C&lpg=PA108&dq=lauritzen%25252520exchangeable&pg=PA108#v=onepage&q=lauritzen%20exchangeable&f=false), which (for me anyway) is the last word on the matter. [Share](https://mathoverflow.net/a/100304 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/ "The current license for this post: CC BY-SA 4.0") Cite [Improve this answer](https://mathoverflow.net/posts/100304/edit) Follow Follow this answer to receive notifications [edited Apr 21, 2023 at 16:02](https://mathoverflow.net/posts/100304/revisions "show all edits to this post") community wiki [3 revs, 2 users 94%](https://mathoverflow.net/posts/100304/revisions "show revision history for this post") [R Hahn](https://mathoverflow.net/users/8719) 3 - 5 There is a recent paper on the arXiv that takes conditional distributions as the foundation of Baysian statistics very seriously: "A categorical foundation for Bayesian probability" by Culbertson and Sturtz, [arxiv.org/abs/1205.1488](http://arxiv.org/abs/1205.1488) Michael Greinecker – [Michael Greinecker](https://mathoverflow.net/users/35357/michael-greinecker) 2012-06-21 22:50:44 +00:00 [Commented Jun 21, 2012 at 22:50](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians#comment257822_100304) - @Michael Greinecker Thanks for the reference, I'll definitely take a look. It isn't my area of research, but I have an abiding interest in keeping up to date on the current thinking about such things. R Hahn – [R Hahn](https://mathoverflow.net/users/8719/r-hahn) 2012-06-21 23:27:08 +00:00 [Commented Jun 21, 2012 at 23:27](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians#comment257832_100304) - 1 \+1 for mentioning Schervish's book. It's the best, imho. (There are a fair number of minor typos, but I think that's excusable for a 700-page first edition.) William DeMeo – [William DeMeo](https://mathoverflow.net/users/9124/william-demeo) 2012-06-22 05:17:26 +00:00 [Commented Jun 22, 2012 at 5:17](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians#comment257862_100304) [Add a comment](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| This answer is useful 3 Save this answer. Show activity on this post. In addition to former proposals, why not the famous two-volume work "Theory of Probability" by Bruno de Finette (In the Wiley classics series) Quotation: "Probability do not exist" A more modern book, with quite another take than the above, is "Bayesian Data Analysis", Third Edition by A Gelman, J B Carlin, H S Stern, D B Dunson, A Vehtari, and D B Rubin (CRC press). This a modern book with emphasis on data analysis and computation. I would say that for today (2017) this is the best starting point. [Share](https://mathoverflow.net/a/100321 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/ "The current license for this post: CC BY-SA 3.0") Cite [Improve this answer](https://mathoverflow.net/posts/100321/edit) Follow Follow this answer to receive notifications [edited Jul 21, 2017 at 9:48](https://mathoverflow.net/posts/100321/revisions "show all edits to this post") community wiki [kjetil b halvorsen](https://mathoverflow.net/posts/100321/revisions "show revision history for this post") [Add a comment](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| This answer is useful 3 Save this answer. Show activity on this post. There is a book of 2018 on the topic, that also presents some recent results of the author: > S. Watanabe, [Mathematical Theory of Bayesian Statistics](https://www.crcpress.com/Mathematical-Theory-of-Bayesian-Statistics/Watanabe/p/book/9781482238068), Chapman and Hall/CRC, Apr 2018 [Share](https://mathoverflow.net/a/306132 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/ "The current license for this post: CC BY-SA 4.0") Cite [Improve this answer](https://mathoverflow.net/posts/306132/edit) Follow Follow this answer to receive notifications answered [Jul 16, 2018 at 14:41](https://mathoverflow.net/posts/306132/revisions "show all edits to this post") community wiki [Dr Fabio Gori](https://mathoverflow.net/posts/306132/revisions "show revision history for this post") 1 - 3 Unlike the provided link, the Amazon page has a preview of the book: [amazon.com/Mathematical-Bayesian-Statistics-Chapman-Monographs/…](https://rads.stackoverflow.com/amzn/click/com/1482238063). The claims in the provided link are vague and bold, mentioning "new statistical laws". Section 9.4 on "phase transitions" is intriguing, but the definition given is very ambiguous. R Hahn – [R Hahn](https://mathoverflow.net/users/8719/r-hahn) 2018-07-16 18:26:50 +00:00 [Commented Jul 16, 2018 at 18:26](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians#comment760968_306132) [Add a comment](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| This answer is useful 3 Save this answer. Show activity on this post. Suggestions: 1. W.M. Bolstad, [Introduction to Bayesian Statistics](http://www.lavoisier.fr/livre/notice.asp?id=OKLWORASRRSOWE) 2nd. Ed., Wiley, 2007. 2. J.M. Bernardo, A.F.M. Smith, [Bayesian Theory](http://books.google.com/books/about/Bayesian_Theory.html?id=9Q9ph-ORI20C&redir_esc=y), Wiley, 2000. 3. J.K. Ghosh, M. Delampady, T. Samanta, [An Introduction to Bayesian Analysis: Theory and Methods](http://books.google.com/books/about/An_Introduction_to_Bayesian_Analysis.html?id=Tf98uAAACAAJ&redir_esc=y), Springer, 2006. **Freely avaliable:** 1. J.F. Kenney, [Mathematics of Statistics Part One](https://archive.org/details/dli.ernet.5933), Chapman & Hall Ltd. 1939. 2. J.F. Kenney, [Mathematics of Statistics Part Two](https://archive.org/details/dli.ernet.5934), Chapman & Hall 1940. [Share](https://mathoverflow.net/a/100298 "Short permalink to this answer") Share a link to this answer Copy link [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/ "The current license for this post: CC BY-SA 4.0") Cite [Improve this answer](https://mathoverflow.net/posts/100298/edit) Follow Follow this answer to receive notifications [edited Apr 25, 2023 at 6:41](https://mathoverflow.net/posts/100298/revisions "show all edits to this post") community wiki [4 revs, 2 users 93%](https://mathoverflow.net/posts/100298/revisions "show revision history for this post") [Papiro](https://mathoverflow.net/users/22714) [Add a comment](https://mathoverflow.net/questions/100288/bayesian-statistics-for-pure-mathematicians "Use comments to ask for more information or suggest improvements. 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Many hold that Bayesian statistics "from a purely mathematical point of view" is entirely coextensive with probability (however it is that you want to define its boundaries as a mathematical discipline). Nonetheless, if I interpret your request as being for a mathematically sophisticated and rigorous exposition on why the Bayesian approach is a worthy one, three book spring to mind. 1. Theory of Statistics by Mark Schervish 2. Bayes Theory by John Hartigan 3. The Bayesian Choice by Christian Robert The first of these is a general graduate text in statistics, but the author gives uncommonly complete coverage of both Bayesian and frequentist methods. The second is a smaller volume and, as I recall, is devoted to some of the more delicate issues surround finite versus countable additivity as relates to using probability distributions as priors in a Bayesian approach. The final book is more general, but the style is more formal than the Bernardo and Smith book mentioned by PaPiro. (This is, in my experience, true of the style of French Bayesians :) As I said, the distinctive elements of the Bayesian perspective are more philosophical than technical, but there are some technical areas that have received attention in the Bayesian community that may be of independent mathematical interest. One would be the role of so-called "improper" priors as mentioned above. Another is the role of conditional distributions as a primitive rather than derived notion, leading to the idea of disintegration, as in this manuscript of [Pollard](http://www.stat.yale.edu/~pollard/Papers/chang-pollard.pdf). Also, because of a keen interest in the application of Monte Carlo methods, Bayesian statisticians have to a lot of work on various aspects of computational methods for sampling from various distributions. Christian Robert is a prominent researcher in this area, and he has a [blog](https://xianblog.wordpress.com/). The current post happens to be about Bayesian foundations. Finally, at the heart of a many arguments in favor of a Bayesian approach (early chapters in Bernardo and Smith and Robert are dedicated to it) are de Finetti type representation theorems, which sanction prior distributions via appeals to exchangeability. You can start with the wiki entry for [de Finetti theorems](https://en.wikipedia.org/wiki/De_Finetti%27s_theorem) and then look at the [work of Persi Diaconis](https://projecteuclid.org/journals/annals-of-probability/volume-8/issue-4/Finite-Exchangeable-Sequences/10.1214/aop/1176994663.full) on the topic. In this vein see also [Lauritzen's monograph](https://books.google.com/books?id=8Pzs_EujC28C&lpg=PA108&dq=lauritzen%25252520exchangeable&pg=PA108#v=onepage&q=lauritzen%20exchangeable&f=false), which (for me anyway) is the last word on the matter.