Monday, December 17, 2012

Galton's Bayesian Machine

Chance magazine has an article by statistician and statistics historian Stephen Stigler on Galton's visualization of Bayes Theorem. He describes Galton's Bayesian machine, likely made up of beads, bins, and glass although the original device doesn't survive. These are, of course, familiar materials for Galton's statistical demonstrations.(This post was intended for some time ago. It got lost in the shuffle).

At the top level, beads are arrayed in deep vertical histogram bins representing the prior distribution, p(θ). A knob is turned and some fall into a bell-shaped horizontal room, representing the likelihood: f(x|θ). The room's back wall bulges away from us placing larger likelihood on central values of θ. If this room were moved more to the right, it would place larger likelihood on larger values of θ. At this second level, those beads falling to the front are retained, inside the bell curve wall, but some beads are rejected falling to the back, outside the wall. This performs the product: f(x|θ) p(θ). Of course at this level some bins are wide and deep, some are very shallow. The knob at this level is then turned and the beads are dropped to the bottom into vertical bins of equal width, rescaling them into a histogram proportional to the posterior distribution: f(θ|x)= f(x|θ) p(θ). Very clever and invented in 1877!

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