Archive for Richard Price

The [errors in the] error of truth [book review]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , on August 10, 2021 by xi'an

OUP sent me this book, The error of truth by Steven Osterling, for review. It is a story about the “astonishing” development of quantitative thinking in the past two centuries. Unfortunately, I found it to be one of the worst books I have read on the history of sciences…

To start with the rather obvious part, I find the scholarship behind the book quite shoddy as the author continuously brings in items of historical tidbits to support his overall narrative and sometimes fills gaps on his own. It often feels like the material comes from Wikipedia, despite expressing a critical view of the on-line encyclopedia. The [long] quote below is presumably the most shocking historical blunder, as the terror era marks the climax of the French Revolution, rather than the last fight of the French monarchy. Robespierre was the head of the Jacobins, the most radical revolutionaries at the time, and one of the Assembly members who voted for the execution of Louis XIV, which took place before the Terror. And later started to eliminate his political opponents, until he found himself on the guillotine!

“The monarchy fought back with almost unimaginable savagery. They ordered French troops to carry out a bloody campaign in which many thousands of protesters were killed. Any peasant even remotely suspected of not supporting the government was brutally killed by the soldiers; many were shot at point-blank range. The crackdown’s most intense period was a horrific ten-month Reign of Terror (“la Terreur”) during which the government guillotined untold masses (some estimates are as high as 5,000) of its own citizens as a means to control them. One of the architects of the Reign of Terror was Maximilien Robespierre, a French nobleman and lifelong politician. He explained the government’s slaughter in unbelievable terms, as “justified terror . . . [and] an emanation of virtue” (quoted in Linton 2006). Slowly, however, over the next few years, the people gained control. In the end, many nobles, including King Louis XVI and his wife Marie-Antoinette, were themselves executed by guillotining”

Obviously, this absolute misinterpretation does not matter (very) much for the (hi)story of quantification (and uncertainty assessment), but it demonstrates a lack of expertise of the author. And sap whatever trust one could have in new details he brings to light (life?). As for instance when stating

“Bayes did a lot of his developmental work while tutoring students in local pubs. He was a respected teacher. Taking advantage of his immediate resources (in his circumstance, a billiard table), he taught his theorem to many.”

which does not sound very plausible. I never heard that Bayes had students  or went to pubs or exposed his result to many before its posthumous publication… Or when Voltaire (who died in 1778) is considered as seventeenth-century precursor of the Enlightenment. Or when John Graunt, true member of the Royal Society, is given as a member of the Académie des Sciences. Or when Quetelet is presented as French and as a student of Laplace.

The maths explanations are also puzzling, from the law of large numbers illustrated by six observations, and wrongly expressed (p.54) as

\bar{X}_n+\mu\qquad\text{when}\qquad n\longrightarrow\infty

to  the Saint-Petersbourg paradox being seen as inverse probability, to a botched description of the central limit theorem  (p.59), including the meaningless equation (p.60)

\gamma_n=\frac{2^{2n}}{\pi}\int_0^\pi~\cos^{2n} t\,\text dt

to de Moivre‘s theorem being given as Taylor’s expansion

f(z)=\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(z-a)^2

and as his derivation of the concept of variance, to another botched depiction of the difference between Bayesian and frequentist statistics, incl. the usual horror

P(68.5<70<71.5)=95%

to independence being presented as a non-linear relation (p.111), to the conspicuous absence of Pythagoras in the regression chapter, to attributing to Gauss the concept of a probability density (when Simpson, Bayes, Laplace used it as well), to another highly confusing verbal explanation of densities, including a potential confusion between different representations of a distribution (Fig. 9.6) and the existence of distributions other than the Gaussian distribution, to another error in writing the Gaussian pdf (p.157),

f(x)=\dfrac{e^{-(z-\mu)^2}\big/2\sigma^2}{\sigma\sqrt{2\pi}}

to yet another error in the item response probability (p.301), and.. to completely missing the distinction between the map and the territory, i.e., the probabilistic model and the real world (“Truth”), which may be the most important shortcoming of the book.

The style is somewhat heavy, with many repetitions about the greatness of the characters involved in the story, and some degree of license in bringing them within the narrative of the book. The historical determinism of this narrative is indeed strong, with a tendency to link characters more than they were, and to make them greater than life. Which is a usual drawback of such books, along with the profuse apologies for presenting a few mathematical formulas!

The overall presentation further has a Victorian and conservative flavour in its adoration of great names, an almost exclusive centering on Western Europe, a patriarchal tone (“It was common for them to assist their husbands in some way or another”, p.44; Marie Curie “agreed to the marriage, believing it would help her keep her laboratory position”, p.283), a defense of the empowerment allowed by the Industrial Revolution and of the positive sides of colonialism and of the Western expansion of the USA, including the invention of Coca Cola as a landmark in the march to Progress!, to the fall of the (communist) Eastern Block being attributed to Ronald Reagan, Karol Wojtyła, and Margaret Thatcher, to the Bell Curve being written by respected professors with solid scholarship, if controversial, to missing the Ottoman Enlightenment and being particularly disparaging about the Middle East, to dismissing Galton’s eugenism as a later year misguided enthusiasm (and side-stepping the issue of Pearson’s and Fisher’s eugenic views),

Another recurrent if minor problem is the poor recording of dates and years when introducing an event or a new character. And the quotes referring to the current edition or translation instead of the original year as, e.g., Bernoulli (1954). Or even better!, Bayes and Price (1963).

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Book Review section in CHANCE.]

an hypothetical chain of transmissions

Posted in Books, Statistics, University life with tags , , , , , , on August 6, 2021 by xi'an

Monte Carlo Markov chains

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , on May 12, 2020 by xi'an

Darren Wraith pointed out this (currently free access) Springer book by Massimiliano Bonamente [whose family name means good spirit in Italian] to me for its use of the unusual Monte Carlo Markov chain rendering of MCMC.  (Google Trend seems to restrict its use to California!) This is a graduate text for physicists, but one could nonetheless expect more rigour in the processing of the topics. Particularly of the Bayesian topics. Here is a pot-pourri of memorable quotes:

“Two major avenues are available for the assignment of probabilities. One is based on the repetition of the experiments a large number of times under the same conditions, and goes under the name of the frequentist or classical method. The other is based on a more theoretical knowledge of the experiment, but without the experimental requirement, and is referred to as the Bayesian approach.”

“The Bayesian probability is assigned based on a quantitative understanding of the nature of the experiment, and in accord with the Kolmogorov axioms. It is sometimes referred to as empirical probability, in recognition of the fact that sometimes the probability of an event is assigned based upon a practical knowledge of the experiment, although without the classical requirement of repeating the experiment for a large number of times. This method is named after the Rev. Thomas Bayes, who pioneered the development of the theory of probability.”

“The likelihood P(B/A) represents the probability of making the measurement B given that the model A is a correct description of the experiment.”

“…a uniform distribution is normally the logical assumption in the absence of other information.”

“The Gaussian distribution can be considered as a special case of the binomial, when the number of tries is sufficiently large.”

“This clearly does not mean that the Poisson distribution has no variance—in that case, it would not be a random variable!”

“The method of moments therefore returns unbiased estimates for the mean and variance of every distribution in the case of a large number of measurements.”

“The great advantage of the Gibbs sampler is the fact that the acceptance is 100 %, since there is no rejection of candidates for the Markov chain, unlike the case of the Metropolis–Hastings algorithm.”

Let me then point out (or just whine about!) the book using “statistical independence” for plain independence, the use of / rather than Jeffreys’ | for conditioning (and sometimes forgetting \ in some LaTeX formulas), the confusion between events and random variables, esp. when computing the posterior distribution, between models and parameter values, the reliance on discrete probability for continuous settings, as in the Markov chain chapter, confusing density and probability, using Mendel’s pea data without mentioning the unlikely fit to the expected values (or, as put more subtly by Fisher (1936), “the data of most, if not all, of the experiments have been falsified so as to agree closely with Mendel’s expectations”), presenting Fisher’s and Anderson’s Iris data [a motive for rejection when George was JASA editor!] as a “a new classic experiment”, mentioning Pearson but not Lee for the data in the 1903 Biometrika paper “On the laws of inheritance in man” (and woman!), and not accounting for the discrete nature of this data in the linear regression chapter, the three page derivation of the Gaussian distribution from a Taylor expansion of the Binomial pmf obtained by differentiating in the integer argument, spending endless pages on deriving standard properties of classical distributions, this appalling mess of adding over the conditioning atoms with no normalisation in a Poisson experiment

P(X=4|\mu=0,1,2) = \sum_{\mu=0}^2 \frac{\mu^4}{4!}\exp\{-\mu\},

botching the proof of the CLT, which is treated before the Law of Large Numbers, restricting maximum likelihood estimation to the Gaussian and Poisson cases and muddling its meaning by discussing unbiasedness, confusing a drifted Poisson random variable with a drift on its parameter, as well as using the pmf of the Poisson to define an area under the curve (Fig. 5.2), sweeping the improperty of a constant prior under the carpet, defining a null hypothesis as a range of values for a summary statistic, no mention of Bayesian perspectives in the hypothesis testing, model comparison, and regression chapters, having one-dimensional case chapters followed by two-dimensional case chapters, reducing model comparison to the use of the Kolmogorov-Smirnov test, processing bootstrap and jackknife in the Monte Carlo chapter without a mention of importance sampling, stating recurrence results without assuming irreducibility, motivating MCMC by the intractability of the evidence, resorting to the term link to designate the current value of a Markov chain, incorporating the need for a prior distribution in a terrible description of the Metropolis-Hastings algorithm, including a discrete proof for its stationarity, spending many pages on early 1990’s MCMC convergence tests rather than discussing the adaptive scaling of proposal distributions, the inclusion of numerical tables [in a 2017 book] and turning Bayes (1763) into Bayes and Price (1763), or Student (1908) into Gosset (1908).

[Usual disclaimer about potential self-plagiarism: this post or an edited version of it could possibly appear later in my Books Review section in CHANCE. Unlikely, though!]

the first Bayesian

Posted in Statistics with tags , , , , , , , on February 20, 2018 by xi'an

In 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.

 

The Richard Price Society

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , on November 26, 2015 by xi'an

As 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.