Bayesian Inference Does Not Lead You Astray... On Average

A decision maker faces an unobserved state of nature. She updates her prior on the state based on the realizations of a signal. In this note, we show that the expected posterior on any given state, taking expectation under the conditional distribution of the signal on this same state, is never lower than the prior on said state. In other words, the expected posterior probability on the true state is never lower than the prior on this state, regardless of what the true state is.