New Metheds for Constructing Monte Carlo Markov
Chains
Jun Liu
Harvard University
We propose a new Metropolis-type transition rule,
which we call the multi-point method. In this rule, one is
allowed to propose multiple correlated (or independent)
trial points and then select a good one among them.
The detailed balance condition is
maintained by using a trick similar to the one suggested in
Frenkel and Smit (1996).
This type of move helps the chain to escape from local energy traps
and can be further combined with the adaptive-direction method of
Gilks, Roberts, and George (1994) to produce more efficient samplers.
If one sees the multi-point method as a generalization of the
Metropolis algorithm, we also have a generalization of the Gibbs sampler.
More precisely, we establish a new theory for iterative conditional sampling
under the setting of transformation group. With this framework,
any Gibbs sampling step can be seen as a random move along the orbit of
a transformation group and the basic principle for guiding the move
is that it leaves the target distribution of interest invariant. In this
way, we can easily generalize the Gibbs sampler to accommodate ``curved"
moves and collective moves, and design more efficient sampling algorithms
(such as the generalized multigrid Monte Carlo).
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