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Given a target distribution $\mu$ and a proposal chain with generator $Q$ ona finite state space, in this paper we study two types of Metropolis-Hastings(MH) generator $M_1(Q,\mu)$ and $M_2(Q,\mu)$ in a continuous-time setting.While $M_1$ is the classical MH generator, we define a new generator $M_2$ thatcaptures the opposite movement of $M_1$ and provide a comprehensive suite ofcomparison results ranging from hitting time and mixing time to asymptoticvariance, large deviations and capacity, which demonstrate that $M_2$ enjoyssuperior mixing properties than $M_1$. To see that $M_1$ and $M_2$ are naturaltransformations, we offer an interesting geometric interpretation of $M_1$,$M_2$ and their convex combinations as $\ell^1$ minimizers between $Q$ and theset of $\mu$-reversible generators, extending the results by Billera andDiaconis (2001). We provide two examples as illustrations. In the first one wegive explicit spectral analysis of $M_1$ and $M_2$ for Metropolised independentsampling, while in the second example we prove a Laplace transform order of thefastest strong stationary time between birth-death $M_1$ and $M_2$.

On hitting time, mixing time and geometric interpretations of Metropolis-Hastings reversiblizations