## Archive for Thomas Bayes

## at the centre of Bayes

Posted in Mountains, pictures, Statistics, Travel, University life with tags 2019 Education Buildings Scotland Awards, Arthur's Seat, Bayes Café, Bayes Centre, Brown University, ECERM, Edinburgh, Holyrood Park, ICMS, jatp, Maxwell Institute Colloquium, Maxwell Institute Graduate School, Scotland, Scottish sun, seminar, Thomas Bayes, University of Edinburgh on October 14, 2019 by xi'an## let the evidence speak [book review]

Posted in Books, Kids, Statistics with tags Bayes grid, Bayesian decision theory, Bayesian inference, book review, court, Durham university, Edwin Jaynes, entropy, evidence, Federalist papers, Thomas Bayes on December 17, 2018 by xi'anThis book by Alan Jessop, professor at the Durham University Business School, aims at presenting Bayesian ideas and methods towards decision making “without formula because they are not necessary; the ability to add and multiply is all that is needed.” The trick is in using a Bayes grid, in other words a two by two table. (There are a few formulas that survived the slaughter, see e.g. on p. 91 the formula for the entropy. Contained in the chapter on information that I find definitely unclear.) When leaving the 2×2 world, things become more complicated and the construction of a prior belief as a probability density gets heroic without the availability of maths formulas. The first part of the paper is about Likelihood, albeit not the likelihood function, despite having the general rule that (p.73)

*belief is proportional to base rate x likelihood*

which is the book‘s version of Bayes’ (base?!) theorem. It then goes on to discuss the less structure nature of prior (or prior beliefs) against likelihood by describing Tony O’Hagan’s way of scaling experts’ beliefs in terms of a Beta distribution. And mentioning Jaynes’ maximum entropy prior without a single formula. What is hard to fathom from the text is how can one derive the likelihood outside surveys. (Using the illustration of 1963 Oswald’s murder by Ruby in the likelihood chapter does not particularly help!) A bit of nitpicking at this stage: the sentence

“The ancient Greeks, and before them the Chinese and the Aztecs…”

is historically incorrect since, while the Chinese empire dates back before the Greek dark ages, the Aztecs only rule Mexico from the 14th century (AD) until the Spaniard invasion. While most of the book sticks with unidimensional parameters, it also discusses more complex structures, for which it relies on Monte Carlo, although the description is rather cryptic (use your spreadsheet!, p.133). The book at this stage turns into a more story-telling mode, by considering for instance the Federalist papers analysis by Mosteller and Wallace. The reader can only follow the process of assessing a document authorship for a single word, as multidimensional cases (for either data or parameters) are out of reach. The same comment applies to the ecology, archeology, and psychology chapters that follow. The intermediary chapter on the “grossly misleading” [Court wording] of the statistical evidence in the Sally Clark prosecution is more accessible in that (again) it relies on a single number. Returning to the ban of Bayes rule in British courts:

In the light of the strong criticism by this court in the 1990s of using Bayes theorem before the jury in cases where there was no reliable statistical evidence, the practice of using a Bayesian approach and likelihood ratios to formulate opinions placed before a jury without that process being disclosed and debated in court is contrary to principles of open justice.

the discussion found in the book is quite moderate and inclusive, in that a Bayesian analysis helps in gathering evidence about a case, but may be misunderstood or misused at the [non-Bayesian] decision level.

In conclusion, Let the Evidence Speak is an interesting introduction to Bayesian thinking, through a simplifying device, the Bayes grid, which seems to come from management, with a large number of examples, if not necessarily all realistic and some side-stories. I doubt this exposure can produce expert practitioners, but it makes for an worthwhile awakening for someone “likely to have read this book because [one] had heard of Bayes but were uncertain what is was” (p.222). With commendable caution and warnings along the way.

## the first Bayesian

Posted in Statistics with tags An Essay towards solving a Problem in the Doctrine of Chances, Bayes' notebooks, Bayesian foundations, David Hume, Richard Price, Royal Society, Statistical Science, Thomas Bayes on February 20, 2018 by xi'an**I**n the first issue of Statistical Science for this year (2018), Stephen Stiegler pursues the origins of Bayesianism as attributable to Richard Price, main author of Bayes’ Essay. (This incidentally relates to an earlier ‘Og piece on that notion!) Steve points out the considerable inputs of Price on this Essay, even though the mathematical advance is very likely to be entirely Bayes’. It may however well be Price who initiated Bayes’ reflections on the matter, towards producing a counter-argument to Hume’s “On Miracles”.

“Price’s caution in addressing the probabilities of hypotheses suggested by data is rare in early literature.”

A section of the paper is about Price’s approach data-determined hypotheses and to the fact that considering such hypotheses cannot easily fit within a Bayesian framework. As stated by Price, “it would be improbable as infinite to one”. Which is a nice way to address the infinite mass prior.

## 10 great ideas about chance [book preview]

Posted in Books, pictures, Statistics, University life with tags Abraham Wald, Alan Turing, Allais' paradox, Alonzo Church, Andrei Kolmogorov, BFF4, book review, Borel-Kolmogorov paradox, Brian Skyrms, Bruno de Finetti, Cardano's formula, CHANCE, David Hume, Dutch book argument, equiprobability, exchangeability, Frank Ramsey, gambling, Gerolamo Cardano, Henri Poincaré, heuristics, Jakob Bernoulli, John Maynard Keynes, John von Neumann, Karl Popper, Martin-Löf, measure theory, p-values, Persi Diaconis, Pierre Simon Laplace, PUP, Radon-Nikodym Theorem, randomness, Richard von Mises, sufficiency, Thomas Bayes, Venn diagram on November 13, 2017 by xi'an*[As I happened to be a reviewer of this book by Persi Diaconis and Brian Skyrms, I had the opportunity (and privilege!) to go through its earlier version. Here are the [edited] comments I sent back to PUP and the authors about this earlier version. All in all, a terrific book!!!]*

**T**he historical introduction (“measurement”) of this book is most interesting, especially its analogy of chance with length. I would have appreciated a connection earlier than Cardano, like some of the Greek philosophers even though I gladly discovered there that Cardano was not only responsible for the closed form solutions to the third degree equation. I would also have liked to see more comments on the vexing issue of *equiprobability*: we all spend (if not waste) hours in the classroom explaining to (or arguing with) students why their solution is not correct. And they sometimes never get it! [And we sometimes get it wrong as well..!] Why is such a simple concept so hard to explicit? In short, but this is nothing but a personal choice, I would have made the chapter more conceptual and less chronologically historical.

“Coherence is again a question of consistent evaluations of a betting arrangement that can be implemented in alternative ways.” (p.46)

The second chapter, about Frank Ramsey, is interesting, if only because it puts this “man of genius” back under the spotlight when he has all but been forgotten. (At least in my circles.) And for joining probability and utility together. And for postulating that probability can be derived from expectations rather than the opposite. Even though betting or gambling has a (negative) stigma in many cultures. At least gambling for money, since most of our actions involve some degree of betting. But not in a rational or reasoned manner. (Of course, this is not a mathematical but rather a psychological objection.) Further, the justification through betting is somewhat tautological in that it assumes probabilities are true probabilities from the start. For instance, the Dutch book example on p.39 produces a gain of .2 only if the probabilities are correct.

> gain=rep(0,1e4) > for (t in 1:1e4){ + p=rexp(3);p=p/sum(p) + gain[t]=(p[1]*(1-.6)+p[2]*(1-.2)+p[3]*(.9-1))/sum(p)} > hist(gain)

As I made it clear at the BFF4 conference last Spring, I now realise I have never really adhered to the Dutch book argument. This may be why I find the chapter somewhat unbalanced with not enough written on utilities and too much on Dutch books.

“The force of accumulating evidence made it less and less plausible to hold that subjective probability is, in general, approximate psychology.” (p.55)

A chapter on “psychology” may come as a surprise, but I feel *a posteriori* that it is appropriate. Most of it is about the Allais paradox. Plus entries on Ellesberg’s distinction between risk and uncertainty, with only the former being quantifiable by “objective” probabilities. And on Tversky’s and Kahneman’s distinction between heuristics, and the framing effect, i.e., how the way propositions are expressed impacts the choice of decision makers. However, it is leaving me unclear about the conclusion that the fact that people behave irrationally should not prevent a reliance on utility theory. Unclear because when taking actions involving other actors their potentially irrational choices should also be taken into account. (This is mostly nitpicking.)

“This is Bernoulli’s swindle. Try to make it precise and it falls apart. The conditional probabilities go in different directions, the desired intervals are of different quantities, and the desired probabilities are different probabilities.” (p.66)

The next chapter (“frequency”) is about Bernoulli’s Law of Large numbers and the stabilisation of frequencies, with von Mises making it the basis of his approach to probability. And Birkhoff’s extension which is capital for the development of stochastic processes. And later for MCMC. I like the notions of “disreputable twin” (p.63) and “Bernoulli’s swindle” about the idea that “chance is frequency”. The authors call the identification of probabilities as limits of frequencies Bernoulli‘s swindle, because it cannot handle zero probability events. With a nice link with the testing fallacy of equating rejection of the null with acceptance of the alternative. And an interesting description as to how Venn perceived the fallacy but could not overcome it: “If Venn’s theory appears to be full of holes, it is to his credit that he saw them himself.” The description of von Mises’ Kollectiven [and the welcome intervention of Abraham Wald] clarifies my previous and partial understanding of the notion, although I am unsure it is that clear for all potential readers. I also appreciate the connection with the very notion of *randomness* which has not yet found I fear a satisfactory definition. This chapter asks more (interesting) questions than it brings answers (to those or others). But enough, this is a brilliant chapter!

“…a random variable, the notion that Kac found mysterious in early expositions of probability theory.” (p.87)

Chapter 5 (“mathematics”) is very important [from my perspective] in that it justifies the necessity to associate measure theory with probability if one wishes to evolve further than urns and dices. To entitle Kolmogorov to posit his axioms of probability. And to define properly conditional probabilities as random variables (as my third students fail to realise). I enjoyed very much reading this chapter, but it may prove difficult to read for readers with no or little background in measure (although some advanced mathematical details have vanished from the published version). Still, this chapter constitutes a strong argument for preserving measure theory courses in graduate programs. As an aside, I find it amazing that mathematicians (even Kac!) had not at first realised the connection between measure theory and probability (p.84), but maybe not so amazing given the difficulty many still have with the notion of conditional probability. (Now, I would have liked to see some description of Borel’s paradox when it is mentioned (p.89).

“Nothing hangs on a flat prior (…) Nothing hangs on a unique quantification of ignorance.” (p.115)

The following chapter (“inverse inference”) is about Thomas Bayes and his posthumous theorem, with an introduction setting the theorem at the centre of the Hume-Price-Bayes triangle. (It is nice that the authors include a picture of the original version of the essay, as the initial title is much more explicit than the published version!) A short coverage, in tune with the fact that Bayes only contributed a twenty-plus paper to the field. And to be logically followed by a second part [formerly another chapter] on Pierre-Simon Laplace, both parts focussing on the selection of prior distributions on the probability of a Binomial (coin tossing) distribution. Emerging into a discussion of the position of statistics within or even outside mathematics. (And the assertion that Fisher was the Einstein of Statistics on p.120 may be disputed by many readers!)

“So it is perfectly legitimate to use Bayes’ mathematics even if we believe that chance does not exist.” (p.124)

The seventh chapter is about Bruno de Finetti with his astounding representation of exchangeable sequences as being mixtures of iid sequences. Defining an implicit prior on the side. While the description sticks to binary events, it gets quickly more advanced with the notion of partial and Markov exchangeability. With the most interesting connection between those exchangeabilities and sufficiency. (I would however disagree with the statement that “Bayes was the father of parametric Bayesian analysis” [p.133] as this is extrapolating too much from the Essay.) My next remark may be non-sensical, but I would have welcomed an entry at the end of the chapter on cases where the exchangeability representation fails, for instance those cases when there is no sufficiency structure to exploit in the model. A bonus to the chapter is a description of Birkhoff’s ergodic theorem “as a generalisation of de Finetti” (p..134-136), plus half a dozen pages of appendices on more technical aspects of de Finetti’s theorem.

“We want random sequences to pass all tests of randomness, with tests being computationally implemented”. (p.151)

The eighth chapter (“algorithmic randomness”) comes (again!) as a surprise as it centres on the character of Per Martin-Löf who is little known in statistics circles. (The chapter starts with a picture of him with the iconic Oberwolfach sculpture in the background.) Martin-Löf’s work concentrates on the notion of randomness, in a mathematical rather than probabilistic sense, and on the algorithmic consequences. I like very much the section on random generators. Including a mention of our old friend RANDU, the 16 planes random generator! This chapter connects with Chapter 4 since von Mises also attempted to define a random sequence. To the point it feels slightly repetitive (for instance Jean Ville is mentioned in rather similar terms in both chapters). Martin-Löf’s central notion is computability, which forces us to visit Turing’s machine. And its role in the undecidability of some logical statements. And Church’s recursive functions. (With a link not exploited here to the notion of probabilistic programming, where one language is actually named Church, after Alonzo Church.) Back to Martin-Löf, (I do not see how his test for randomness can be implemented on a real machine as the whole test requires going through the entire sequence: since this notion connects with von Mises’ Kollektivs, I am missing the point!) And then Kolmororov is brought back with his own notion of complexity (which is also Chaitin’s and Solomonov’s). Overall this is a pretty hard chapter both because of the notions it introduces and because I do not feel it is completely conclusive about the notion(s) of randomness. A side remark about casino hustlers and their “exploitation” of weak random generators: I believe Jeff Rosenthal has a similar if maybe simpler story in his book about Canadian lotteries.

“Does quantum mechanics need a different notion of probability? We think not.” (p.180)

The penultimate chapter is about Boltzmann and the notion of “physical chance”. Or statistical physics. A story that involves Zermelo and Poincaré, And Gibbs, Maxwell and the Ehrenfests. The discussion focus on the definition of probability in a thermodynamic setting, opposing time frequencies to space frequencies. Which requires ergodicity and hence Birkhoff [no surprise, this is about ergodicity!] as well as von Neumann. This reaches a point where conjectures in the theory are yet open. What I always (if presumably naïvely) find fascinating in this topic is the fact that ergodicity operates without requiring randomness. Dynamical systems can enjoy ergodic theorem, while being completely deterministic.) This chapter also discusses quantum mechanics, which main tenet requires probability. Which needs to be defined, from a frequency or a subjective perspective. And the Bernoulli shift that brings us back to random generators. The authors briefly mention the Einstein-Podolsky-Rosen paradox, which sounds more metaphysical than mathematical in my opinion, although they get to great details to explain Bell’s conclusion that quantum theory leads to a mathematical impossibility (but they lost me along the way). Except that we “are left with quantum probabilities” (p.183). And the chapter leaves me still uncertain as to why statistical mechanics carries the label *statistical*. As it does not seem to involve inference at all.

“If you don’t like calling these ignorance priors on the ground that they may be sharply peaked, call them nondogmatic priors or skeptical priors, because these priors are quite in the spirit of ancient skepticism.” (p.199)

And then the last chapter (“induction”) brings us back to Hume and the 18th Century, where somehow “everything” [including statistics] started! Except that Hume’s strong scepticism (or skepticism) makes induction seemingly impossible. (A perspective with which I agree to some extent, if not to Keynes’ extreme version, when considering for instance financial time series as stationary. And a reason why I do not see the criticisms contained in the Black Swan as pertinent because they savage normality while accepting stationarity.) The chapter rediscusses Bayes’ and Laplace’s contributions to inference as well, challenging Hume’s conclusion of the impossibility to finer. Even though the representation of ignorance is not unique (p.199). And the authors call again for de Finetti’s representation theorem as bypassing the issue of whether or not there is such a thing as chance. And escaping inductive scepticism. (The section about Goodman’s grue hypothesis is somewhat distracting, maybe because I have always found it quite artificial and based on a linguistic pun rather than a logical contradiction.) The part about (Richard) Jeffrey is quite new to me but ends up quite abruptly! Similarly about Popper and his exclusion of induction. From this chapter, I appreciated very much the section on skeptical priors and its analysis from a meta-probabilist perspective.

There is no conclusion to the book, but to end up with a chapter on induction seems quite appropriate. (But there is an appendix as a probability tutorial, mentioning Monte Carlo resolutions. Plus notes on all chapters. And a commented bibliography.) Definitely recommended!

*[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE. As appropriate for a book about Chance!]*

## The Seven Pillars of Statistical Wisdom [book review]

Posted in Books, pictures, Statistics, University life with tags Boston, Florence Nightingale, foundations, JSM 2014, Karl Pearson, likelihood, Pierre Simon Laplace, quincunx, R.A. Fisher, residuals, Stephen Stigler, T.E. Lawrence, The Seven Pillars of Statistical Wisdom, Thomas Bayes, trek rule theorem, W. Gosset on June 10, 2017 by xi'an**I** remember quite well attending the ASA Presidential address of Stephen Stigler at JSM 2014, Boston, on the seven pillars of statistical wisdom. In connection with T.E. Lawrence’s 1926 book. Itself in connection with Proverbs IX:1. Unfortunately wrongly translated as *seven pillars* rather than *seven sages*.

As pointed out in the Acknowledgements section, the book came prior to the address by several years. I found it immensely enjoyable, first for putting the field in a (historical and) coherent perspective through those seven pillars, second for exposing new facts and curios about the history of statistics, third because of a literary style one would wish to see more often in scholarly texts and of a most pleasant design (and the list of reasons could go on for quite a while, one being the several references to Jorge Luis Borges!). But the main reason is to highlight the unified nature of Statistics and the reasons why it does not constitute a subfield of either Mathematics or Computer Science. In these days where centrifugal forces threaten to split the field into seven or more disciplines, the message is welcome and urgent.

Here are Stephen’s pillars (some comments being already there in the post I wrote after the address):

*aggregation*, which leads to gain information by throwing away information, aka the sufficiency principle. One (of several) remarkable story in this section is the attempt by Francis Galton, never lacking in imagination, to visualise the average man or woman by superimposing the pictures of several people of a given group. In 1870!*information*accumulating at the √n rate, aka precision of statistical estimates, aka CLT confidence [quoting de Moivre at the core of this discovery]. Another nice story is Newton’s wardenship of the English Mint, with musing about [his] potential exploiting this concentration to cheat the Mint and remain undetected!*likelihood*as the right calibration of the amount of information brought by a dataset [including Bayes’ essay as an answer to Hume and Laplace’s tests] and by Fisher in possible the most impressive single-handed advance in our field;*intercomparison*[i.e. scaling procedures from variability within the data, sample variation], from Student’s [a.k.a., Gosset‘s] t-test, better understood and advertised by Fisher than by the author, and eventually leading to the bootstrap;*regression*[linked with Darwin’s evolution of species, albeit paradoxically, as Darwin claimed to have faith in nothing but the irrelevant Rule of Three, a challenging consequence of this theory being an unobserved increase in trait variability across generations] exposed by Darwin’s cousin Galton [with a detailed and exhilarating entry on the quincunx!] as conditional expectation, hence as a true Bayesian tool, the Bayesian approach being more specifically addressed in (on?) this pillar;*design of experiments*[re-enters Fisher, with his revolutionary vision of changing all factors in Latin square designs], with an fascinating insert on the 18th Century French Loterie, which by 1811, i.e., during the Napoleonic wars, provided 4% of the national budget!;*residuals*which again relate to Darwin, Laplace, but also Yule’s first multiple regression (in 1899), Fisher’s introduction of parametric models, and Pearson’s χ² test. Plus Nightingale’s diagrams that never cease to impress me.

The conclusion of the book revisits the seven pillars to ascertain the nature and potential need for an eight pillar. It is somewhat pessimistic, *at least my reading of it was*, as it cannot (and presumably does not want to) produce any direction about this new pillar and hence about the capacity of the field of statistics to handle in-coming challenges and competition. With some amount of exaggeration (!) I do hope the analogy of the seven pillars that raises in me the image of the beautiful ruins of a Greek temple atop a Sicilian hill, in the setting sun, with little known about its original purpose, remains a mere analogy and does not extend to predict the future of the field! By its very nature, this wonderful book is about foundations of Statistics and therefore much more set in the past and on past advances than on the present, but those foundations need to move, grow, and be nurtured if the field is not to become a field of ruins, a methodology of the past!

## The Richard Price Society

Posted in Books, pictures, Statistics, Travel, University life with tags ISBA, Richard Price, Richard Price Society, Thomas Bayes, Wales, Welsh on November 26, 2015 by xi'an **A**s an item of news coming to me via ISBA News, I learned of the Richard Price Society and of its endeavour to lobby for the Welsh government to purchase Richard Price‘s birthplace as an historical landmark. As discussed in a previous post, Price contributed so much to Bayes’ paper that one may wonder who made the major contribution. While I am not very much inclined in turning old buildings into museums, feel free to contact the Richard Price Society to support this action! Or to sign the petition there. Which I cannot resist but reproduce in Welsh:

## Datblygwch Fferm Tynton yn Ganolfan Ymwelwyr a Gwybodaeth

Rydym yn galw ar Lywodraeth Cymru i gydnabod cyfraniad pwysig Dr Richard Price nid yn unig i’r Oes Oleuedig yn y ddeunawfed ganrif, ond hefyd i’r broses o greu’r byd modern yr ydym yn byw ynddo heddiw, a datblygu ei fan geni a chartref ei blentyndod yn ganolfan wybodaeth i ymwelwyr lle gall pobl o bob cenedl ac oed ddarganfod sut mae ei gyfraniadau sylweddol i ddiwinyddiaeth, mathemateg ac athroniaeth wedi dylanwadu ar y byd modern.