To compute an expectation E[ g y] using MCMC simulation, we rewrite (8.149) to integrate over all values of ? ? as

where we introduce the indicator function

We then define the sampling density

such that

We use MCMC simulation to generate a sequence ( ? ^{[ m ]}, ? ^{[ m ]}) that is distributed according to ? _s( ?, ? y), such that for a large number N _s of samples, the expectation is approximately

The error of this approximation is normally distributed with a standard deviation proportional to if the samples are independent. To generate this sequence, we use the Metropolis algorithm, known in statistics as Metropolis Hastings sampling. While other MC algorithms (e.g. Gibbs sampling) may yield superior performance, here we use only the Metropolis algorithm, which we have encountered already in Chapter 7. From the current state ( ? ^{[ m ]}, ? ^{[ m ]}), we propose a move ( ? ^{[ m ]}, ? ^{[ m ]}) ? ( ? ^(new), ? ^(new)) and then decide whether to accept this new state as the next member ( ? ^{[ m +1]}, ? ^{[ m +1]}) of the sequence. We generate a new trial state by displacing at random either ? or ?. For some specified fraction of ? moves f _? ?