Summarizing posterior distributions with maxima

We have seen that writing a mathematical expression for a posterior distribution is relatively easy. The name of the inference game is making sense of this expression. Sampling via Markov chain Monte Carlo is a powerful way to do that, but can have prohibitive computational cost.

To get a much cruder picture of a posterior distribution, we can find the parameter values that maximize the posterior PDF (or, rarely, PMF). We can then locally approximate the posterior as Normal. Such a summary is far inferior to what we can obtain with MCMC, but is useful when practical considerations become important. In fact, optimization-based techniques, where we attempt to summarize a posterior distribution with its maximum, are by far the most widely used approaches.