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

Michael C. H. Choi () and Lu-Jing Huang ()
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Michael C. H. Choi: The Chinese University of Hong Kong
Lu-Jing Huang: Fujian Normal University

Journal of Theoretical Probability, 2020, vol. 33, issue 2, 1144-1163

Abstract: Abstract Given a target distribution $$\mu $$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 $$M_1(Q,\mu )$$M1(Q,μ) and $$M_2(Q,\mu )$$M2(Q,μ) in a continuous-time setting. While $$M_1$$M1 is the classical MH generator, we define a new generator $$M_2$$M2 that captures the opposite movement of $$M_1$$M1 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 $$M_2$$M2 enjoys superior mixing properties than $$M_1$$M1. To see that $$M_1$$M1 and $$M_2$$M2 are natural transformations, we offer an interesting geometric interpretation of $$M_1$$M1, $$M_2$$M2 and their convex combinations as $$\ell ^1$$ℓ1 minimizers between Q and the set of $$\mu $$μ-reversible generators, extending the results by Billera and Diaconis (Stat Sci 16(4):335–339, 2001). We provide two examples as illustrations. In the first one, we give explicit spectral analysis of $$M_1$$M1 and $$M_2$$M2 for Metropolized independent sampling, while in the second example, we prove a Laplace transform order of the fastest strong stationary time between birth–death $$M_1$$M1 and $$M_2$$M2.

Keywords: Markov chains; Metropolis–Hastings algorithm; Additive reversiblization; Hitting time; Mixing time; Asymptotic variance; Large deviations; 60J27; 60J28 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-019-00903-2

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