Metropolis-Hastings Algorithm

M-H Sampler

Fact Gibbs sampler requires simulation from the full conditional distributions for all . As before these may not be available. Instead of sampling from , we can sample from where is easy to sample from, then decide whether to accept sample as next point in Markov chain simulation, or to stay at the current point.

Algorithm Metropolis-Hastings Sampler For any proposed distribution such that the Markov chain is still -irreducible and aperiodic:

1. Initialize parameter vector . Set .
2. Generate proposed point
3. Generate . If wherethen accept . Else .
4. Increment and repeat the above.

Thrm Detailed Balance Suppose is a transition density for a given . Then a that satisfies the reversibility condition:has as its invariant distribution.

M-H Special Cases

Def Random Walk Metropolis Sampler If for some arbitrary density then the M-H algorithm is a random walk aswe call it random walk metropolis sampler.

Prop For random walk metropolis sampler, we have