# A Free Course Book on Bayesian Inference: [6.] The Rule of Succession and The Problem of Points

Since 2017, Dora Matzke and I have been teaching the master course “Bayesian Inference for Psychological Science”. Over the years, the syllabus for this course matured into a book (and an accompanying book of answers) titled “Bayesian inference from the ground up: The theory of common sense”. The current plan is to finish the book in the next few months, and share the pdf publicly online. For now, you can click here or on the cover page below to read the first 182 pages. Today we are adding another three chapters: “The Rule of Succession”, “The Problem of Points”, and the appendix chapter “Pascal’s Arithmetical Triangle”.

Highlights:

1. In “The Rule of Succession”,  we discuss the “beta prediction rule”; When a binomial chance parameter θ has a beta(a,b) distribution, the probability that the next trial will be a success equals the mean of the distribution, that is, a/(a+b). This makes it easy to derive a number of Laplacean predictions.
2. For a long time, we were under the impression that “the verification of a consequence renders a coinjecture more credible” as Pólya put it. Every black raven you encounter ought to provide evidence for the proposition that “all ravens are black”. Appendix B to “The Rule of Succession” provides eight counterexamples that should disabuse anybody from this notion. In 1972, Paul Berent published a forgotten one-page article, “Disconfirmation by positive instances”, that absolutely nails it. Take-home message: background knowledge matters.
3. In “The Problem of Points”, we showcase the JASP implementation (in the LearnBayes module) and emphasize the distinction between a game of chance (that features only aleatory uncertainty) and a game of skill (that also features epistemic uncertainty). Interestingly, by adding epistemic uncertainty you should become more confident that the player in the lead will win the match, and you should therefore accord them a larger stake.
4. In the appendix chapter “Pascal’s Arithmetical Triangle”, we explain the Galton board and Pascal’s triangle. We are particularly happy with Viktor Beekman’s drawings.

References

Berent, P. (1972). Disconfirmation by positive instances. Philosophy of Science, 39, 522.

Wagenmakers, E.-J., & Matzke, D. (in preparation). Bayesian inference from the ground up: The theory of common sense.

Wagenmakers, E.-J., & Matzke, D. (in preparation). Bayesian inference from the ground up: Common sense in practice.