## 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.

## Clockers [book review]

Posted in Books, Travel with tags , , , , , , , , , on March 15, 2014 by xi'an

Throughout my recent trip to Canada, I read bits and pieces of Clockers by Richard Price and I finished reading it last Sunday. It is an impressive piece of literature and I am surprised I was not aware of its existence until amazon.com suggested it to me (as I was checking for recent books by another Richard, Richard Morgan!). Guessing from the summary it could be of interest and from comments it was sort of a classic, I ordered it more or less on a whim (given a comfortable balance on my amazon.com account, thanks to ‘Og’s readers!) It took me a few pages to realise the plot was deeply set in the 1990’s, not only because this was the high of the crack epidemics, but also since the characters (drug dealers and policemen) therein are all using beepers, instead of cellphones, and street phone booths).

“It’s like a math problem. Juan got whacked at point X, he drove away losing blood at the rate of a pint every ninety seconds. He was driving forty-five miles an hour and he bought the farm two miles inside the tunnel (…) So for ten points, [who] in what New Jersey town did Juan?” Clockers (p.272)

The plot of Clockers is vaguely a detective story as an aging and depressed homicide officer, Rosso, hunts the murderer of a drug dealer, being convinced from the start that the self-declared murderer Victor did not do it. In parallel, and somewhat more closely, the book follows the miserable plight and thoughts and desires of Victor’s brother, Strike, who is head of a local crack dealing network, under the domination of the charismatic and berserk Rodney Little… But the resolution of the crime matters very little, much less than the exposure of the deadly economics of the drug traffic in inner cities (years before Freakonomics!), of the constant fight of single mothers to bring food and structure to their dysfunctional families, to the widespread recourse to moonlighting, and above all to the almost physical impossibility to escape one’s environment (even for smart and decent kids like Victor and, paradoxically enough, the drug-dealing Strike) by lack of prospect and exposure to anything or anywhere else, as well as social pressure, early pregnancies and gang-related micro-partitioning of cities.

When I mentioned Clockers to Andrew, he told me that he also liked it very much but that the characters were not quite “real”. I somewhat agree in that, while the economics, the sociology and the practice of drug-dealing sound very accurately reproduced (for all I know!), the characters are more caricaturesque or picturesque than natural. The stomach disease of Strike sounds too much like an allegory of both his schizophrenic split between running the drug trade and looking for a definitive quit, while the sacrifice of his brother makes little sense, except as a form either of suicide or of escape from an environment he can no longer stand. What is most surprising is that Richard Price (just like Michael Crichton) is  a practised screenwriter (who collaborated to Spike Lee’s 1995 Clockers). So he knows how to run an efficient story with convincing characters and plot(s). Hence my little theory of a picaresque novel… (Here is Jim Shepard’s enthusiastic review of Clockers. With the definitely accurate title of “Sympathy for the dealer”.)

## Bayes 250th versus Bayes 2.5.0.

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , on July 20, 2013 by xi'an

More than a year ago Michael Sørensen (2013 EMS Chair) and Fabrizzio Ruggeri (then ISBA President) kindly offered me to deliver the memorial lecture on Thomas Bayes at the 2013 European Meeting of Statisticians, which takes place in Budapest today and the following week. I gladly accepted, although with some worries at having to cover a much wider range of the field rather than my own research topic. And then set to work on the slides in the past week, borrowing from my most “historical” lectures on Jeffreys and Keynes, my reply to Spanos, as well as getting a little help from my nonparametric friends (yes, I do have nonparametric friends!). Here is the result, providing a partial (meaning both incomplete and biased) vision of the field.

Since my talk is on Thursday, and because the talk is sponsored by ISBA, hence representing its members, please feel free to comment and suggest changes or additions as I can still incorporate them into the slides… (Warning, I purposefully kept some slides out to preserve the most surprising entry for the talk on Thursday!)