The Metropolis-Hastings Update

The Metropolis-Hastings update (Metropolis, et al., 1953; Hastings, 1970) simulates a distribution specified by an unnormalized density h with respect to a measure . It uses an auxiliary `proposal' density with respect to having the following properties

• must be normalized, .
• q(x, y) can be evaluated for all x and y.
• for each x, it is possible to simulate realizations from the distribution having density with respect to .

Other than this there is no particular relation between q and h. There is always an infinite variety of proposal densities that will do the job.

The Metropolis-Hastings update changes the state x as follows

1. Simulate .
2. Evaluate the `Hastings ratio'

   

3. (`Metropolis rejection') Move to y with probability . Otherwise stay at x.

Note that `Metropolis rejection' has no relation to the `rejection sampling' of ordinary independent-sample Monte Carlo. If step 3 `rejects' the `proposal' y and stays at the current state x, then the Markov chain stays at x for two consecutive iterations (at least it does so if this update is not composed with another update). This is not the case in ordinary rejection sampling, which keeps making proposals until one is accepted and the next sample is the first accepted proposal.