Monday, May 5, 2014

A simple identity for Bayesian updating

For random variables A and X, consider the relationship
E{XA}=E{X}E{A}+ ρXAσXσA
which, up to a bit of arithmetic, is basically the definition of correlation. If A is a binary variable, though, we can do more with this; among other things, in this case σA2=E{A}(1-E{A}). Conflating the variable A a bit with the "event" A=1, and doing a bit of algebra, we get
The effect of the arrival of new information on the expected value of a variable is proportional to the square root of the odds ratio. Among other things, it can't be more than σX times the square root of the odds ratio, though this bound, which (obviously?) is reached when X is a linear function of A and therefore is a binary variable, can be more directly derived in that context.

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