Does additional data always reduce posterior variance?

A discussion over lunch today brought up the fact that additional data does not always decrease the size of a confidence interval. This post will look at this from a Bayesian perspective. In general, new information reduces your uncertainty regarding whatever you’re estimating. The posterior distribution becomes more concentrated as more data are collected. That’s what happens “in general” but does it necessarily happen every time you get new data? Conceivably if you get surprising data, data that is very unlikely given your current prior, posterior uncertainty might increase. To show that this is the case, suppose the probability of success in some binary trial has parameter θ and that θ has a beta prior. You could imagine this prior to be the posterior after having made some number of previous observations. Can a new observation increase the posterior variance in θ? If so, under what conditions? The variance of a beta(a, b) random variable is ab / (a + b)²(a + b + 1). After…