Multiple-try Metropolis
Liu et al. (2000) have suggested a modified MH algorithm, which they call the Multiple-try Metropolis algorithm (MTM), which allows larger step sizes whilst still retaining a reasonable acceptance rate.
Suppose is an arbitrary proposal function. We require that only if . Additionally, is the likelihood function.
Define where is a non-negative symmetric function in and that can be chosen by the user.
Now suppose the current state is . The MTM algorithm is as follows:
1) Draw k independent trial proposals from . Compute the weights for each of these.
2) Select from the with probability proportional to the weights.
3) Now produce a reference set by drawing from the distribution . Set (the current point).
4) Accept with probability
It can be shown that this method satisfies the detailed balance property and therefore produces a reversible Markov chain with as the stationary distribution.
If is symmetric (as is the case for the multivariate normal distribution), then one can choose which gives
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