Assessing significance in a Markov chain without mixing

Abstract

We present a statistical test to detect that a presented state of a reversible Markov chain was not chosen from a stationary distribution. In particular, given a value function for the states of the Markov chain, we would like to show rigorously that the presented state is an outlier with respect to the values, by establishing a p value under the null hypothesis that it was chosen from a stationary distribution of the chain. A simple heuristic used in practice is to sample ranks of states from long random trajectories on the Markov chain and compare these with the rank of the presented state; if the presented state is a 0.1% outlier compared with the sampled ranks (its rank is in the bottom 0.1% of sampled ranks), then this observation should correspond to a p value of 0.001. This significance is not rigorous, however, without good bounds on the mixing time of the Markov chain. Our test is the following: Given the presented state in the Markov chain, take a random walk from the presented state for any number of steps. We prove that observing that the presented state is an ε-outlier on the walk is significant at p = √ 2ε under the null hypothesis that the state was chosen from a stationary distribution. We assume nothing about the Markov chain beyond reversibility and show that significance at p ≈ √ ε is best possible in general. We illustrate the use of our test with a potential application to the rigorous detection of gerrymandering in Congressional districting.Markov chain | mixing time | gerrymandering | outlier | p value T he essential problem in statistics is to bound the probability of a surprising observation under a null hypothesis that observations are being drawn from some unbiased probability distribution. This calculation can fail to be straightforward for a number of reasons. On the one hand, defining the way in which the outcome is surprising requires care; for example, intricate techniques have been developed to allow sophisticated analysis of cases where multiple hypotheses are being tested. On the other hand, the correct choice of the unbiased distribution implied by the null hypothesis is often not immediately clear; classical tools like the t test are often applied by making simplifying assumptions about the distribution in such cases. If the distribution is well-defined but is not be amenable to mathematical analysis, a p value can still be calculated using bootstrapping if test samples can be drawn from the distribution.A third way for p value calculations to be nontrivial occurs when the observation is surprising in a simple way and the null hypothesis distribution is known but where there is no simple algorithm to draw samples from this distribution. In these cases, the best candidate method to sample from the null hypothesis is often through a Markov chain, which essentially takes a long random walk on the possible values of the distribution. Under suitable conditions, theorems are available that guarantee that the chain converges to its stationary distribution, allowing a random sample t...

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