Galton & simulation

September 27, 2010
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Stephen Stigler has written a paper in the Journal of the Royal Statistical Society Series A on Francis Galton’s analysis of (his cousin) Charles Darwin’ Origin of Species, leading to nothing less than Bayesian analysis and accept-reject algorithms!

“On September 10th, 1885, Francis Galton ushered in a new era of Statistical Enlightenment with an address to the British Association for the Advancement of Science in Aberdeen. In the process of solving a puzzle that had lain dormant in Darwin’s Origin of Species, Galton introduced multivariate analysis and paved the way towards modern Bayesian statistics. The background to this work is recounted, including the recognition of a failed attempt by Galton in 1877 as providing the first use of a rejection sampling algorithm for the simulation of a posterior distribution, and the first appearance of a proper Bayesian analysis for the normal distribution.”

The point of interest is that Galton proposes through his (multi-stage) quincunx apparatus a way to simulate from the posterior of a normal mean (here is an R link to the original quincunx). This quincunx has a vertical screen at the second level that acts as a way to physically incorporate the likelihood (it also translates the fact that the likelihood is in another “orthogonal” space, compared  with the prior!):

“Take another look at Galton’s discarded 1877 model for natural selection (Fig. 6). It is nothing less that a workable simulation algorithm for taking a normal prior (the top level) and a normal likelihood (the natural selection vertical screen) and finding a normal posterior (the lower level, including the rescaling as a probability density with the thin front compartment of uniform thickness).”

Besides a simulation machinery (steampunk Monte Carlo?!), it also offers the enormous appeal of proposing the derivation of the normal-normal posterior for the very first time:

“Galton was not thinking in explicit Bayesian terms, of course, but mathematically he has posterior $mathcal{N}(0,C_2)proptomathcal{N}(0,A_2)times f(x=0|y)$. This may be the earliest appearance of this calculation; the now standard derivation of a posterior distribution in a normal setting with a proper normal prior. Galton gave the general version of this result as part of his 1885 development, but the 1877 version can be seen as an algorithm employing rejection sampling that could be used for the generation of values from a posterior distribution. If we replace $f(x)$ above by the density $mathcal{N}(a,B_2)$, his algorithm would generate the posterior distribution of Y given X=a, namely $mathcal{N}(aC_2/B_2, C_2)$. The assumption of normality is of course needed for the particular formulae here, but as an algorithm the normality is not essential; posterior values for any prior and any location parameter likelihood could in principle be generated by extending this algorithm.”

This historical entry is furthermore interesting at another (anecdotal) level in that it is reminding me of a visit I made as a teenager to the Birmingham museum of Sciences where I saw a quincunx for the first time and got fairly intrigued by the stable law exhibited therein… (In a completely unrelated manner, let me point out the excellent pastiche of Dickensian dark novels called The Quincunx by Charles Palliser.)

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