Chapter 5 Bayesian Inference

We shall all be bayesian in 2020.

Dennis Lindley in the preface of (Finetti 1974, ix).

For (DeGroot and Schervish 2012, 378), ‘statistical inference is a procedure that produces a probabilistic statement about some or all parts of a statistical model’. Conclusions obtained from the data support inductive reasoning, i.e., that which starts from the particular case to the general case and which is opposed to deductive reasoning which goes from the general to the particular case. The inductive principle fits into the notion of a decision-maker inferring about universal/population parameters from a sample (particular case).

Bayesian inference has its origins in the posthumous article by (Bayes 1763), communicated by his friend Richard Price. The derivations of Bayes’ ideas are extensive and deep mathematically and philosophically, discussed by great names in Science in countless books, articles and compilations over these more than 250 years. Therefore, it is understood that the best approach for this material is to indicate the state-of-the-art of Bayesian application considering established references. \(\\\)

Exercise 5.1 Watch the videos The Bayesian Trap from the channel Veritasium and Bayes theorem, the geometry of changing beliefs from the channel 3Blue1Brown. Remember that you can activate the subtitles (CC button) and change the language in the settings (gear icon > subtitles).

Exercise 5.2 Read the article When Did Bayesian Inference Become “Bayesian”?) by (Fienberg 2006). \(\\\)

One of the main reasons for recent advances in Bayesian statistics research is the increasing ease of access to computational resources, both hardware and software. In the R language there are many libraries for Bayesian application. The CRAN Task View ²⁰ of Bayesian inference provides an up-to-date compendium of packages related to the subject. The software JASP presented in Section 1.1.6 has a menu that mixes classical and Bayesian methods. The software Stan presented in Section 1.1.10 can be considered the state-of-the-art in Bayesian computing.

Exercise 5.3 Indicate possible parameters obtained from
a. a poem
b. a song
c. a photography
d. a painting
e. a book
f. a musical album
g. film

Example 5.1 (Little Cultural Moment) Inference is a live album by pianist Marilyn Crispell and saxophonist Tim Berne, recorded at the Toronto Jazz Festival in 1992 and released by the Music & Arts label. The theme song can be heard here.

References

Bayes, Thomas. 1763. “An Essay Towards Solving a Problem in the Doctrine of Chances. By the Late Rev. Mr. Bayes, FRS Communicated by Mr. Price, in a Letter to John Canton, AMFR S.” Philosophical Transactions of the Royal Society of London, no. 53: 370–418. https://www.ias.ac.in/article/fulltext/reso/008/04/0080-0088.

DeGroot, Morris H, and Mark J Schervish. 2012. Probability and Statistics. Pearson Education.

Fienberg, Stephen E. 2006. “When did Bayesian inference become" Bayesian"?” Bayesian Analysis 1 (1): 1–40. https://projecteuclid.org/journals/bayesian-analysis/volume-1/issue-1/When-did-Bayesian-inference-become-Bayesian/10.1214/06-BA101.pdf.

———. 1974. “Theory of Probability. A Critical Introductory Treatment, 2 Volumes. Translation by a. Machi and a.f.m. Smith of 1970 Book.” Wiley ISBN.

According to the official R documentation, CRAN Task Views are intended to provide some guidance about which packages in CRAN are relevant to tasks related to a particular topic. They provide a brief overview of included packages and are intended to be sharply focused so that it is sufficiently clear which packages should be included (or excluded) - and are not intended to endorse the “best” packages for a given task.↩︎