9.1 Introduction to numerical methods
9.1.1 Monte Carlo methods
Bayesian statistics proceeds smoothly and easily as long as we stick to well-known distributions and use conjugate priors. But in real life, it is often difficult to model the situations which arise in practice in that way; instead, we often arrive at a posterior
but have no easy way of finding the constant multiple and have to have recourse to some numerical technique. This is even more true when we come to evaluate the marginal density of a single component 
of 
. A simple example was given earlier in Section 3.9 on ‘The circular normal distribution’, but the purpose of this chapter is to discuss more advanced numerical techniques and the Gibbs sampler in particular.
There are, of course, many techniques available for numerical integration, good discussions of which can be found in, for example, Davis and Rabinowitz (1984), Evans (1993) or Evans and Swartz (2000). Most of the methods we shall discuss are related to the idea of ‘Monte Carlo’ integration as a method of finding an expectation. In the simplest version of this, we write
where the points x1, x2, … , xn are chosen as independent ‘pseudo-random’ numbers with ...
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