Appendix A. Inconsistency of posterior mean when prior depends on sample size.
As an illustration of the consequences of having priors depend on sample size, consider the case of a Binomial random variable X consisting of N independent Bernoulli trials with success parameter p. Suppose p has a beta prior distribution with parameters a and b, denoted . The posterior distribution of p is with mean
this is a weighted average of the prior mean and the maximum likelihood estimator The important feature is that as , the weight on the prior mean goes to zero, provided that a and b are fixed.
Now suppose that is fixed, but that (a + b) = kN, for a fixed value of k. Then the posterior mean becomes
The weight on the prior mean does not go to zero as . Indeed, the posterior mean converges to something different than the MLE, hence is not consistent.