11 Sampling Methods
For most probabilistic models of practical interest, exact inference is intractable, and
so we have to resort to some form of approximation. In Chapter 10, we discussed
inference algorithms based on deterministic approximations, which include methods
such as variational Bayes and expectation propagation. Here we consider approxi-
mate inference methods based on numerical sampling, also known asMonte Carlo
techniques.
Although for some applications the posterior distribution over unobserved vari-
ables will be of direct interest in itself, for most situations the posterior distribution
is required primarily for the purpose of evaluating expectations, for example in order
to make predictions. The fundamental problem that we therefore wish to address in
this chapter involves finding the expectation of some functionf(z)with respect to a
probability distributionp(z). Here, the components ofzmight comprise discrete or
continuous variables or some combination of the two. Thus in the case of continuous