Comparison Inequalities and Fastest-Mixing Markov Chains

Details Comparison Inequalities and Fastest-Mixing Markov Chains Comparison Inequalities and Fastest-Mixing Markov Chains

2011-09-28

arxiv.org/abs/1109.6075v2

Mathematics

Probability

the original submission was accepted subject to minor revision by Annals of Applied Probability; this second posting of the ar

Scientific paper

We introduce a new partial order on the class of stochastically monotone (SM) Markov kernels having a given stationary distribution pi on a given finite partially ordered state space X. When K precedes L in this partial order we say that K and L satisfy a comparison inequality. We establish that if K_1, ..., K_t and L_1, ..., L_t are SM and reversible and K_s precedes L_s for s = 1, ..., t, then K_1 ... K_t precedes L_1 ... L_t. In particular, in the time-homogeneous case we have that K^t precedes L^t for every t if K and L are SM and reversible and K precedes L, and using this we show that (for suitable common initial distributions) the Markov chain Y with kernel K mixes faster than the chain Z with kernel L, in the strong sense that at every time t the discrepancy -- measured by total variation distance or separation or L^2-distance -- between the law of Y_t and pi is smaller than that between the law of Z_t and pi. Using comparison inequalities together with specialized arguments to remove the stochastic monotonicity restriction, we answer a question of Persi Diaconis by showing that, among all symmetric birth-and-death kernels on the path X = {0, ..., n}, the one (we call it the uniform chain) that produces fastest convergence from initial state 0 to the uniform distribution has transition probability 1/2 in each direction along each edge of the path, with holding probability 1/2 at each endpoint. We also use comparison inequalities (i) to identify, when pi is a given log-concave distribution on the path, the fastest-mixing stochastically monotone birth-and-death chain started at 0, and (ii) to recover and extend a result of Peres and Winkler that extra updates do not delay mixing for monotone spin systems. Among the fastest-mixing chains in (i), we show that the chain for uniform pi is slowest in the sense of maximizing separation at every time.

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