Reversal of Markov Chains and the Forget Time

Abstract

We study three quantities that can each be viewed as the time needed for a finite irreducible Markov chain to 'forget' where it started. One of these is the mixing time, the minimum mean length of a stopping rule that yields the stationary distribution from the worst starting state. A second is the forget time, the minimum mean length of any stopping rule that yields the same distribution from any starting state. The third is the reset time, the minimum expected time between independent samples from the stationary distribution. Our main results state that the mixing time of a chain is equal to the mixing time of the time-reversed chain, while the forget time of a chain is equal to the reset time of the reverse chain. In particular, the forget time and the reset time of a time-reversible chain are equal. Moreover, the mixing time lies between absolute constant multiples of the sum of the forget time and the reset time. We also derive an explicit formula for the forget time, in terms of the 'access times' introduced in [11]. This enables us to relate the forget and reset times to other mixing measures of the chain.