Comparison Inequalities and Fastest-mixing Markov Chains

We introduce a new partial order on the class of stochastically monotone Markov kernels having a given stationary distribution π on a given finite partially ordered state space X . When K L in this partial order we say that K and L satisfy a comparison inequality. We establish that if K1, . . . ,Kt and L1, . . . , Lt are reversible and Ks Ls for s = 1, . . . , t, then K1 · · ·Kt L1 · · ·Lt. In particular, in the time-homogeneous case we have K L for every t if K and L are reversible and K 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-distance—between the law of Yt and π is smaller than that between the law of Zt and π. 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 π is a given log-concave distribution on the path, the fastestmixing 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 π is slowest in the sense of maximizing separation at every time.