Mixing Time Estimation in Reversible Markov Chains from a Single Sample Path
NIPS, page 1459--1467. (September 2015)Abstract
This article provides the first procedure for computing a fully
data-dependent interval that traps the mixing time t_mix of a finite
reversible ergodic Markov chain at a prescribed confidence level. The
interval is computed from a single finite-length sample path from the
Markov chain, and does not require the knowledge of any parameters of
the chain. This stands in contrast to previous approaches, which
either only provide point estimates, or require a reset mechanism, or
additional prior knowledge.
The interval is constructed around the relaxation time
t_relax, which is strongly related to the mixing time, and
the width of the interval converges to zero roughly
at a root-n rate, where n is the length of the sample path.
Upper and lower bounds are given on the number of samples required to
achieve constant-factor multiplicative accuracy. The lower bounds
indicate that, unless further restrictions are placed on the chain, no
procedure can achieve this accuracy level before seeing each state at
least Omega(t_relax) times on the average. Finally, future
directions of research are identified.