Journal
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
Volume 50, Issue 5, Pages 2743-2762Publisher
SIAM PUBLICATIONS
DOI: 10.1137/110832288
Keywords
Markov chain; Markov chain Monte Carlo; asymptotic variance; average-case analysis; worst-case analysis; rate of convergence; reversibility; nonreversibility
Categories
Funding
- National Science Council [98-2115-M-001-007-MY3, 99-2115-M-305-001, 100-2115-M-001-006]
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Let chi be a finite space and let pi be an underlying probability on chi. For any real-valued function f defined on chi, we are interested in calculating the expectation of f under pi. Let X-0, X-1,..., X-n,... be a Markov chain generated by some transition matrix P with invariant distribution pi. The time average, 1/n Sigma(n-1)(k=0) f(X-k), is a reasonable approximation to the expectation, E-pi[f(X)]. Which matrix P minimizes the asymptotic variance of 1/n Sigma(n-1)(k=0) f(X-k)? The answer depends on f. Rather than a worst-case analysis, we will identify the set of P's that minimize the average asymptotic variance, averaged with respect to a uniform distribution on f.
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