Highlights:

- As detailed in the chapter on Buffon’s needle, there is much, much more to Buffon than the needle. Buffon constructed a working version of the “Archimedes death ray”, wrote a best-selling encyclopedia of 44 bulky volumes, largely constructed what is now the fabulous “Jardin des Plantes” in Paris, and ran one of Europe’s most modern forges. Hidden in the base of one of the statues of Buffon (not the one shown above) is….his
*cerebellum*. - I could talk more about the pancake chapter, but I think anything I’ll say will fall flat after the statement that Buffon’s statue contains his cerebellum. You’ll have to check out the chapter yourself!

**References**

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.

]]>Highlights:

- 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.
- 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.
- 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.
- 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.

]]>In “Learning from the likelihood ratio”, we discuss the scenario of two point hypotheses, which is arguably the simplest inferential scenario: “Miruna comes home and discovers that it’s Dutch pancakes for dinner. Hurray! She knows the pancakes were baked by either of her parents, Andy and Bobbie, but she does not know which one. The only clue as to the identity of the baker is provided by the composition of the pancakes: Andy has a probability of producing a bacon pancake of θA = 0.40, whereas that probability is θB = 0.80 for Bobbie.”

In “An infinite number of hypotheses”, we first increase the number of candidate bakers from 2 to 11; next, we discuss the pancake proclivity of Mr. X, whose value of θ can take on infinitely many values. It is only here that we introduce the beta distribution. In many introductory texts, the beta distribution is the point of departure. However, the beta distribution and the associated results are not so trivial:

“It might seem, indeed, utterly impossible to calculate out a problem having an

infinite number of hypotheses, but the wonderful resources of the integral

calculus enable this to be done (…) But I may add that though the integral

calculus is employed as a means of summing infinitely numerous results, we in

no way abandon the principles of combinations already treated.” (Jevons, 1874)

**References**

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.

]]>My personal highlights from “Measuring probability”:

- Viktor Beekman’s drawing of Ramsey’s farmer (“Harriet is not 100% certain about the direction of her hotel. Her degree of uncertainty can be measured by the distance she is just willing to walk in order to obtain the correct information from a friendly Frisian farmer.”)
- A rare photo of a young Dennis Lindley, courtesy of Janet, Rowan, and Robert Lindley.

As far as the chapter “Coherence” goes, I believe it is an underappreciated topic: “coherence is akin to good health; it is usually enjoyed without much thought. Only when it breaks down does it suddenly become apparent that it was in fact crucial all along.” The chapter features Aristotle (of course), Poincaré, Pólya, de Finetti, and, of course, Dennis Lindley. One provocative opinion is summarized in the box below:

**References**

Probability is an odd mathematical discipline, as even the greatest scientific minds can struggle with seemingly simple problems. Sometimes, brilliant scholars will analyze a problem in probability theory, arguing that probability is difficult and the results are counterintuitive, provinding elaborate explanations of their proposed solution…and still get it wrong. I am increasingly convinced that the only way to reduce the chance of falling prey to paradoxes in probability is to religiously stick to Bayes’ rule — that is, by manipulating mathematical symbols instead of relying on Venn diagrams or probability trees.

**References**

Is the universe deterministic? The question has zero practical relevance, but at the same time one cannot help but wonder whether there is any other question worth studying Determinism is also an interesting theme running through the Matrix movies. Consider for instance the following dialogue:

*Merovingian*: “It is, of course, the way of all things. You see, there is only one constant, one universal, it is the only real truth: causality. Action–reaction; cause–and effect.”

*Morpheus*: “Everything begins with choice.”

*Merovingian*: “No. Wrong. Choice is an illusion, created between those with power, and those without.(…) This is the nature of the universe. We struggle against it, we fight to deny it, but it is of course pretense, it is a lie. Beneath our poised appearance, the truth is we are completely out of control. Causality. There is no escape from it, we are forever slaves to it. Our only hope, our only peace is to understand it, to understand the ‘why’.” [The Merovingian stands up from the table]
*Persephone*: “Where are you going?”

*Merovingian*: “Please, ma cherie, I’ve told you, we are all victims of causality. I drink too much wine, I must take a piss. Cause and effect. Au revoir.”

It seems that the Merovingian was an avid student of Schopenhauer, who, in 1841, made a similar claim, with comparible gusto:

“You can

dowhat youwill: but at each given moment of your life you canwillonly one determined thing and by no means anything other than this one. (…)So, throughout this ever increasing heterogeneity, incommensurability and unintelligibility of the relation between cause and effect, has the

necessityit presupposes also decreased at all? In no way, not in the slightest. As necessarily as the rolling ball sets the one at rest in motion, so too must the Leyden flask discharge itself when touched by the other hand, so must arsenic kill any living thing, so must the seed grain that was stored dry and showed no alteration through millennia germinate, grow and develop into a plant as soon as it is placed in the appropriate soil and exposed to the influences of air, light, heat and moisture. The cause is more complicated, the effect more heterogeneous, but the necessity with which it occurs is not one hair’s breadth smaller. (…)It is definitely neither metaphor nor hyperbole, but a quite dry and literal truth, that just as a ball cannot start into motion on a billiard table until it receives an impact, no more can a human being stand up from his chair until a motive draws or drives him away: but then his standing up is as necessary and inevitable as the ball’s rolling after the impact. (…)

Under presupposition of free will each human action would be an inexplicable miracle — an effect without cause. (…)

Everything that happens, from the greatest to the smallest, happens necessarily. Whatever happens, necessarily happens. Whoever is alarmed at these propositions still has some things to learn and others to unlearn: but after that he will recognize that they are the most abundant source of comfort and relief. — Our deeds are truly no first beginning, and so in them nothing really new attains existence: ratherthrough what we do, we merely come to experience what we are. (…)Wishing that some incident had not happened is a foolish self-torment: for it means wishing something absolutely impossible, and is as irrational as the wish that the sun should rise in the West. Because every happening, great or small, occurs

strictlynecessarily, it is totally vain to reflect on how trivial and accidental were the causes that brought about that incident and how very easily they could have been different. For this is illusory, in that they all occurred with just as strict a necessity and had their effect with just as much power as those in consequence of which the sun rises in the East. Rather we ought to regard events as they occur with the same eye as the print that we read, knowing full well that it stood there before we read it.”

**References**

In the preface, we outline five ways in which the book sets itself apart from most other introductions to Bayesian inference:

- Our introduction is “from the ground up”, which means it is geared towards students in a field such as psychology. It it decidedly
*not*an introduction for mathematics professors at MIT (although some might like the book regardless). - The book aligns itself closely with the philosophy of Sir Harold Jeffreys, and therefore it makes a clear distinction between parameter estimation and hypothesis testing. And when we conduct hypothesis tests, we use the dreaded Bayes factor (and explain why this is such a good idea).
- Throughout the book, we focus on prediction and repeat the mantra that underlies all Bayesian learning: “
*accounts of the world that predicted observed data successfully enjoy a boost in plausibility, whereas accounts that predicted poorly suffer a decline*“. - “The fourth way in which this book stands out is that we stress historical development. The heroes of this book include Pierre-Simon Laplace (1749–1827), Augustus De Morgan (1806–1871), William Stanley Jevons (1835–1882), Henri Poincaré (1854–1912), J. B. S. Haldane (1892–1964), Dorothy Maud Wrinch (1894–1976), and of course Sir Harold Jeffreys (1891–1989). Many chapters provide abundant historical background and elaborate quotations. Some students have told us that long quotations are boring. We heap scorn on this notion. Our heroes may no longer be around to do a Ted Talk or record a TikTok video, but their

words have lost none of their eloquence, relevance, and vision.” - The book takes full advantages of JASP, and in particular the “Learn Bayes” module.

We hope you’ll enjoy this taster. There’s a lot more to come.

**References**

Overgrown with moss and generally in a sorry state, this is hardly a worthy monument to the man who pioneered the way in which causes can be inferred from consequences. This is all the more surprising because ISBA, the International Society for Bayesian Analysis, has a fund that is dedicated to the upkeep of the tomb. According to the ISBA website, the “Thomas Bayes Tomb Maintenance Fund” is a “a small fund, replenished by member donations, devoted towards continuing the upkeep of the tomb of the Reverand Thomas Bayes, namesake of ISBA, into the future. Initial contributions to the fund were from BEST (Bayesian Efficient Strategic Trading) LLC of Hoboken, NJ, after an introduction by ISBA.” The accompanying photo on the ISBA website shows the tomb in a relatively good state:

The initial restoration was apparently carried out in 2007, and this webpage shows the before and after photos. Immediately after its restoration, the tomb looked impressive, all shiny and bright white. It seems that by 2023, the tomb has regrettably regressed to its pre-2007 state. How sad! ISBA, can we please fix this?!

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