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2015 ; 36
(3
): 917-941
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SHARP ENTRYWISE PERTURBATION BOUNDS FOR MARKOV CHAINS
#MMPMID26491218
Thiede E
; VAN Koten B
; Weare J
SIAM J Matrix Anal Appl
2015[]; 36
(3
): 917-941
PMID26491218
show ga
For many Markov chains of practical interest, the invariant distribution is
extremely sensitive to perturbations of some entries of the transition matrix,
but insensitive to others; we give an example of such a chain, motivated by a
problem in computational statistical physics. We have derived perturbation bounds
on the relative error of the invariant distribution that reveal these variations
in sensitivity. Our bounds are sharp, we do not impose any structural assumptions
on the transition matrix or on the perturbation, and computing the bounds has the
same complexity as computing the invariant distribution or computing other bounds
in the literature. Moreover, our bounds have a simple interpretation in terms of
hitting times, which can be used to draw intuitive but rigorous conclusions about
the sensitivity of a chain to various types of perturbations.