How to Combine Fast Heuristic Markov Chain Monte Carlo with Slow Exact Sampling
Antar Bandyopadhyay, University of California, Berkeley
David J. Aldous, University of California, Berkeley
Abstract
Given a probability law $pi$ on a set S and a function
$g : S rightarrow R$, suppose one wants to estimate the mean
$bar{g} = int g dpi$. The Markov Chain Monte Carlo method
consists of inventing and simulating a Markov chain with stationary
distribution $pi$. Typically one has no a priori bounds on the chain's
mixing time, so even if simulations suggest rapid mixing one cannot
infer rigorous confidence intervals for $bar{g}$.
But suppose there is also a separate method which (slowly) gives samples
exactly from $pi$.
Using n exact samples, one could
immediately get a confidence interval of length
O(n-1/2). But one can do better. Use each exact sample as the initial
state of a Markov chain, and run each of these n chains for m
steps. We show how to construct confidence intervals which are always
valid, and which, if the (unknown) relaxation time of the chain
is sufficiently small relative to m/n, have length
O(n-1 log n) with high probability.
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