Chapter 6

Simulation and Monte Carlo Integration

This chapter addresses the simulation of random draws **X**_{1}, . . ., **X**_{n} from a target distribution *f*. The most frequent use of such draws is to perform *Monte Carlo integration*, which is the statistical estimation of the value of an integral using evaluations of an integrand at a set of points drawn randomly from a distribution with support over the range of integration [49].

Estimation of integrals via Monte Carlo simulation can be useful in a wide variety of settings. In Bayesian analyses, posterior moments can be written in the form of an integral but typically cannot be evaluated analytically. Posterior probabilities can also be written as the expectation of an indicator function with respect to the posterior. The calculation of risk in Bayesian decision theory relies on integration. Integration is also an important component in frequentist likelihood analyses. For example, marginalization of a joint density relies upon integration. Example 5.1 illustrates an integration problem arising from the maximum likelihood fit of a generalized linear mixed model. A variety of other integration problems are discussed here and in Chapter 7.

Aside from its application to Monte Carlo integration, simulation of random draws from a target density *f* is important in many other contexts. Indeed, Chapter 7 is devoted to a specific strategy for Monte Carlo integration called Markov chain Monte Carlo. Bootstrap methods, stochastic search algorithms, ...