Assessing significance in a Markov chain without mixing

作者：Chikina Maria; Frieze Alan; Pegden Wesley^*

来源：Proceedings of the National Academy of Sciences of the United States of America, 2017, 114(11): 2860-2864.

DOI：10.1073/pnas.1617540114

• Markov chain
• mixing time
• gerrymandering
• outlier
• p value

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 Rho 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 Rho 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 epsilon-outlier on the walk is significant at Rho = root 2 epsilon 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 Rho approximate to root epsilon 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.

• 出版日期2017-3-14