Eigentime identity for asymmetric finite Markov chains

Cui, Hao; Mao, Yong-Hua

doi:10.1007/s11464-010-0067-8

Cited by 20 publications

(21 citation statements)

References 12 publications

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“…where the right hand side is the Kemeny's constant which is independent of the starting state i. We refer interested readers to Cui and Mao (2010); Kirkland (2014); Levene and Loizou (2002); Mao (2004); Pitman and Tang (2018) for further references on this parameter. • Forest representation of mean hitting time: Let G(P ) be the weighted direct graph on vertices X and arc weights to be the corresponding transition probabilities.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On resistance distance of Markov chain and its sum rules

Choi

2019

Linear Algebra and its Applications

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Motivated by the notion of resistance distance on graph, we define a new resistance distance between two states on a given finite ergodic Markov chain based on its fundamental matrix. We prove a few equivalent formulations and discuss its relation with other parameters of the Markov chain such as its group inverse, stationary distribution, eigenvalues or hitting time. In addition, building upon existing sum rules for the hitting time of Markov chain, we give sum rules of this new resistance distance of Markov chains that resembles the sum rules of the resistance distance on graph. This yields Markov chain counterparts of various classical formulae such as Foster's first formula or the Kirchhoff index formulae.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On resistance distance of Markov chain and its sum rules

Choi

2019

Linear Algebra and its Applications

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“…where τ A := inf {t; X t ∈ A} is the first hitting time of A by X, and as usual we write τ j = τ {j} . Note that for uniformly ergodic Markov chain, the eigentime identity Aldous and Fill (2002); Cui and Mao (2010); Mao (2004) is given by…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Hitting time and mixing time bounds of Stein’s factors

Choi¹

2018

Electron. Commun. Probab.

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For any discrete target distribution, we exploit the connection between Markov chains and Stein's method via the generator approach and express the solution of Stein's equation in terms of expected hitting time. This yields new upper bounds of Stein's factors in terms of the parameters of the Markov chain, such as mixing time and the gradient of expected hitting time. We compare the performance of these bounds with those in the literature, and in particular we consider Stein's method for discrete uniform, binomial, geometric and hypergeometric distribution. As another application, the same methodology applies to bound expected hitting time via Stein's factors. This article highlights the interplay between Stein's method, modern Markov chain theory and classical fluctuation theory.

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“…In fact, t av equals to the sum of the reciprocals of the non-zero eigenvalues of −Q, and it also has close connection with the notion of strong ergodicity, see for example Cui and Mao (2010); Mao (2004). • (Total variation mixing time) For > 0, we write the total variation mixing time t mix ( ) to be…”

Section: Geometric Interpretation Of M 1 and Mmentioning

confidence: 99%

On Hitting Time, Mixing Time and Geometric Interpretations of Metropolis–Hastings Reversiblizations

Choi

Huang

2019

J Theor Probab

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Given a target distribution µ and a proposal chain with generator Q on a finite state space, in this paper we study two types of Metropolis-Hastings (MH) generator M 1 (Q, µ) and M 2 (Q, µ) in a continuous-time setting. While M 1 is the classical MH generator, we define a new generator M 2 that captures the opposite movement of M 1 and provide a comprehensive suite of comparison results ranging from hitting time and mixing time to asymptotic variance, large deviations and capacity, which demonstrate that M 2 enjoys superior mixing properties than M 1 . To see that M 1 and M 2 are natural transformations, we offer an interesting geometric interpretation of M 1 , M 2 and their convex combinations as 1 minimizers between Q and the set of µ-reversible generators, extending the results by Billera and Diaconis (2001). We provide two examples as illustrations. In the first one we give explicit spectral analysis of M 1 and M 2 for Metropolised independent sampling, while in the second example we prove a Laplace transform order of the fastest strong stationary time between birth-death M 1 and M 2 .

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Eigentime identity for asymmetric finite Markov chains

Abstract: Two kinds of eigentime identity for asymmetric finite Markov chains are proved both in the ergodic case and the transient case.

Cited by 20 publications

References 12 publications

On resistance distance of Markov chain and its sum rules

On resistance distance of Markov chain and its sum rules

Hitting time and mixing time bounds of Stein’s factors

On Hitting Time, Mixing Time and Geometric Interpretations of Metropolis–Hastings Reversiblizations

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