Monte Carlo Methods

Monte Carlo Methods

In a typical Bayesian analysis, the computations involved in eliminating the missing parameters and other unknown quantities are so difficult that one has to resort to some numerical approaches to complete the required summations and integrations. Among all the numerical approaches, Monte Carlo methods are perhaps the most versatile, flexible, and powerful [275].

Suppose that we can generate random samples (either independent or dependent)

from the joint posterior distribution (8.2); or we can generate random samples

from the marginal posterior distribution p(X|Y). Then we can approximate the marginal posterior p(x_i|Y) by the empirical distribution (i.e., the histogram) based on the corresponding component in the Monte Carlo sample (i.e., x⁽¹⁾_i, x⁽²⁾_i,...x⁽ⁿ⁾_i) and approximate the inference (8.4) by

Equation 8.11

As noted in Section 8.1, most Monte Carlo techniques fall into one of the following two categories: Markov chain Monte Carlo methods, corresponding to batch processing, and sequential Monte Carlo methods, corresponding to adaptive processing. These are discussed in the remainder of this chapter.