On the optimal transition matrix for markov chain monte carlo sampling

Let X be a finite space and let π be an underlying probability on X. For any real-valued function f defined on X, we are interested in calculating the expectation of f under π. Let X ₀,X ₁, . . .,X _n, . . . be a Markov chain generated by some transition matrix P with invariant distribution p. The time average, 1/n σ ^n-1 _k=0 f(X _k), is a reasonable approximation to the expectation, Eπ[f(X)]. Which matrix P minimizes the asymptotic variance of 1/n σ ^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.