Computing the nearest reversible Markov chain

2015 ◽  
Vol 22 (3) ◽  
pp. 483-499 ◽  
Author(s):  
A. J. N. Nielsen ◽  
M. Weber
Keyword(s):  
Markov Chain ◽  
Reversible Markov Chain

scholarly journals Assessing significance in a Markov chain without mixing

2017 ◽  
Vol 114 (11) ◽  
pp. 2860-2864 ◽  
Author(s):  
Maria Chikina ◽  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Stationary Distribution ◽  
Potential Application ◽  
Null Hypothesis ◽  
Value Function ◽  
Mixing Time ◽  
Statistical Test ◽  
Reversible Markov Chain ◽  
A 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 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.

ON THE SPECTRAL GAP OF A TIME REVERSIBLE MARKOV CHAIN

1999 ◽  
Vol 13 (1) ◽  
pp. 95-101 ◽  
Author(s):  
Olivier François
Keyword(s):  
Markov Chain ◽  
Spectral Gap ◽  
Reversible Markov Chain

We give a new bound on the spectral gap of an ergodic time reversible Markov chain in term of the conductance of the chain.

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