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

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.