## the worst possible proof [X’ed]

Posted in Books, Kids, Statistics, University life with tags , , , , , , on July 18, 2015 by xi'an

Another surreal experience thanks to X validated! A user of the forum recently asked for an explanation of the above proof in Lynch’s (2007) book, Introduction to Applied Bayesian Statistics and Estimation for Social Scientists. No wonder this user was puzzled: the explanation makes no sense outside the univariate case… It is hard to fathom why on Earth the author would resort to this convoluted approach to conclude about the posterior conditional distribution being a normal centred at the least square estimate and with σ²X’X as precision matrix. Presumably, he has a poor opinion of the degree of matrix algebra numeracy of his readers [and thus should abstain from establishing the result]. As it seems unrealistic to postulate that the author is himself confused about matrix algebra, given his MSc in Statistics [the footnote ² seen above after “appropriately” acknowledges that “technically we cannot divide by” the matrix, but it goes on to suggest multiplying the numerator by the matrix

$(X^\text{T}X)^{-1} (X^\text{T}X)$

which does not make sense either, unless one introduces the trace tr(.) operator, presumably out of reach for most readers]. And this part of the explanation is unnecessarily confusing in that a basic matrix manipulation leads to the result. Or even simpler, a reference to Pythagoras’  theorem.

## Le Monde puzzle [#869]

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , on June 8, 2014 by xi'an

An uninteresting Le Monde mathematical puzzle:

Solve the system of equations

• a+b+c=16,
• b+c+d=12,
• d+c+e=16,
• e+c+f=18,
• g+c+a=15

for 7 different integers 1≤a,…,g9.

Indeed, the final four equations determine d=a-4, e=b+4, f=a-2, g=b-1 as functions of a and b. While forcing 5≤a, 2b≤5, and  7a+b≤15. Hence, 5 possible values for a and 4 for b. Which makes 20 possible solutions for the system. However the fact that a,b,c,d,e,f,g are all different reduces considerably the possibilities. For instance, b must be less than a-4. The elimination of impossible cases leads in the end to consider b=a-5 and b=a-7. And eventually to a=8, b=3… Not so uninteresting then. A variant of Sudoku, with open questions like what is the collection of the possible values of the five sums, i.e. of the values with one and only one existing solution? Are there cases where four equations only suffice to determine a,b,c,d,e,f,g?

Apart from this integer programming exercise, a few items of relevance in this Le Monde Science & Medicine leaflet.  A description of the day of a social sciences worker in front of a computer, in connection with a sociology (or sociometry) blog and a conference on Big Data in sociology at Collège de France. A tribune by the physicist Marco on data sharing (and not-sharing) illustrated by an experiment on dark matter called Cogent. And then a long interview of Matthieu Ricard, who argues about the “scientifically proven impact of meditation”, a sad illustration of the ease with which religions permeate the scientific debate [or at least the science section of Le Monde] and mingle scientific terms with religious concepts (e.g., the fusion term of “contemplative sciences”). [As another “of those coincidences”, on the same day I read this leaflet, Matthieu Ricard was the topic of one question on a radio quizz.]

## Correlations between the physical and social sciences

Posted in Books, Statistics, University life with tags , , , , , , , on January 18, 2012 by xi'an

This is probably the most bizarre book I have received for review (so far).  Its title is wide-ranging: Correlations Between the Physical and Social Sciences.  Its cover is enticing: a picture of the young Albert Einstein. Its purpose is wide:

The thesis of this monograph is that societies in general are governed by objective laws that have their roots in human nature. The task of the social scientist is to discover and explore those laws (…) Null hypotheses and alternative rival hypotheses developed by social scientists must eclectically correlated to mathematical formulae or the laws of physics in order to advance non-speculative, unbiased knowledge.” V.J. Belfiglio (p.x)

So the thesis advanced in Correlations Between the Physical and Social Sciences by Valentine Belfiglio is that social problems can be represented in terms of physical laws. The 41 pages book pushes this argument through four cases studies.

The first case study relates marital assimilation of minority groups into dominate core cultures with Graham’s Law for the diffusion of gases. The second case study relates the mutual hostility of political leaders with the Mirror Equation employed in basic geometric optics. The third case study relates the duration of major American military conflicts to the formulae for empirical and subjective probabilities. The fourth case study relates the radioactive decay formula for radioactive substances to the rate of decline of several extinct empires” V.J. Belfiglio (p.xi)

As the author himself recognises, “the four case studies in this monograph do not provide definitive answers.” My opinion is that they do not provide answers at all! Indeed, the first chapter contains two 2×2 tables about the endogamous preferences of Mexican and Italian inhabitants of Dallas, Texas. A chi-square test concludes that Mexicans prefer endogamy and that Italians do not. Although Graham’s Law is re-expressed there as “marital assimilation being inversely proportional to the square root of the population densities” (p.3), there is no result based on the data supporting this law. The second chapter is trying to “explore the mutuality of hostility between the Bush and Ahmandinejad (sic) administrations. Spearman’s Rho correlation coefficient” (p.11) is used and found to demonstrate “a perfect positive correlation” (p.12), although the data is quantitative (intensity of hostility between 1 and 9) and not paired. (The study simply shows that the empirical cdfs of the hostility values for both sides are approximately the same, Spearman’s rho test being inappropriate there.) The connection with optics is at best tenuous. Chapter 3 centres on a table for the durations of major American (meaning US) military conflicts. A mere observation is that the US “has been engaged in major wars 56.5 percent of the time between 1775-2010.” (p.24) but Valentine Belfiglio turns this into “empirical probability” (i.e the frequency of wars), a “subjective probability” (i.e. the average number of years of peace between wars), and the “number of possible interaction channels” (i.e. a combination number) as a way to link American foreign policy with probability theory. Again, the connection is non-existent. The fourth and final chapter is about the “correlation between the decay of radioactive substances and the rate of decline of empires.” (p.31) The data is made of the duration of seven empires, associated with estimates of their half-life. The paper concludes on “a perfect negative correlation between the half-lives of empires and their rates of decline” (p.35), which is not very surprising when considering that one is a monotonic function of the other…

I conclude with the words of Henry Wadsworth Longfellow: “Sometimes we may learn more from a man’s errors, than from his virtues”.” V.J. Belfiglio (p.40)

There is therefore not much to discuss about this book: it does not go beyond stating the obvious, while the connection between the observed social phenomena and generic physical laws remains at the level of a literary ellipse, not of a scientific demonstration. I am deeply puzzled at why a publisher would want to publish this… Any review of the material should have shown the author was out of his depth—his speciality at Texas Woman’s University is Government—in this particular endeavour of proving that “mathematical formulae and the law of physics can take scholars further in deriving conclusions from sets of assumptions than can inferential statistics” (back-cover).