Comparison inequalities and fastest-mixing Markov chains

Abstract

We introduce a new partial order on the class of stochastically monotone Markov kernels having a given stationary distribution $\pi$ on a given finite partially ordered state space ${ \mathcal{X}}$. When $K\preceq L$ in this partial order we say that $K$ and $L$ satisfy a comparison inequality. We establish that if $K_{1},\ldots,K_{t}$ and $L_{1},\ldots,L_{t}$ are reversible and $K_{s}\preceq L_{s}$ for $s=1,\ldots,t$, then $K_{1}\cdots K_{t}\preceq L_{1}\cdots L_{t}$. In particular, in the time-homogeneous case we have $K^{t}\preceq L^{t}$ for every $t$ if $K$ and $L$ are reversible and $K\preceq L$, 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 ${ \mathcal{X}}=\{0,\ldots,n\}$, 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.