Yale University
Department of Statistics
Seminar

Monday, October 9, 2000
New Methods for Constructing Monte Carlo Markov Chains
Jun Liu
Department of Statistics
Harvard University

Seminar to be held in Room 107, 24 Hillhouse at 4:15 pm


New Methods for Constructing Monte Carlo Markov Chains
We propose a new Metropolis-type transition rule, which we call the {\it 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 algithms (such as the generalized multigrid Monte Carlo).