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A few words about Thomas Bayes






P(H/D) = [P(D/H) x P(H)] ÷ P(L)


Thomas Bayes, a mid-eighteenth century English mathematician, dead since 1761, performed magic the ramifications of which continue to unfold, and in directions of interest to the world of finance today. But I won't really get into the applications here. I'll just state the main point associated with this man, so I can refer you back to this post if I eventually do get to them. I want this blog to be a bit sticky.

Probability

In Bayes’ day, the study of probability was mostly about issues of this form: if we know a certain underlying fact, what is the probability that we will make certain observations? If we know there are 100 blue balls and 40 red balls in the urn, and the urn has been thoroughly shaken, what is the probability that we will draw two blue balls in succession from the urn?

Bayes’ famous posthumous manuscript was about “inverse probability” in that it inverted this problem. If we know only the observed data, what can we infer about the underlying matters of fact? If we have just drawn two blue balls from the urn, what can we infer about the mixture of balls inside?

That ms itself didn’t lay down what is nowadays known as Bayes’ theorem, but it took some big steps in that direction, steps further developed at the start of the nineteenth century by the astronomer Pierre-Simon Laplace into the present accepted formulations. (One common formulation of Bayes' theorem is portrayed in large characters at the  top of this blog entry, where H stands for "hypothesis" and D stands for the supporting data.)

From these starts, Bayesians have also developed the idea of a Bayesian game: that is, a game in which a player has to try to figure out the characteristics of other players despite incomplete information on the basis of various signals. People are more complicated than urns filled with balls, but otherwise this is a natural development of the underlying idea. Think of a poker player trying to decide whether the other guy’s facial tick means that he has lousy cards, or just that he has a facial tic.
The probability that he has lousy cards given the tic is equal to the probability of a tic if he has lousy cards, multiplied by the probability that he has lousy cards in general, all divided by the probability of facial tics.
 
Or (to dip our toe into finance after all) think of a plot of land that may or may not have oil beneath it, and how its value might rise when the market learns of geological features that could be a sign of such oil.


I think I have that right....

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