Accelerating reversible Markov chains

Non-reversible Metropolis-Hastings

“…Roberts and Rosenthal (2004)]. The following result is obtained in Sun et al (2010), for a more extensive argument see Chen and Hwang (2013). Then for all f : S → R, we have σ 2 f,P ≤ σ 2 f,K , and there exists an f such that σ 2 f,P < σ 2 f,K .…”

“…This can be shown experimentally in special cases Suwa and Todo (2010), Turitsyn et al (2011), Vucelja (2014), theoretically in special cases Diaconis et al (2000), Neal (2004), and in fact, also in general Sun et al (2010), Chen and Hwang (2013), with respect to asymptotic variance. See also Rey-Bellet and Spiliopoulos (2014) for improved asymptotic variance of non-reversible diffusion processes on compact manifolds.…”

On the rate of convergence of the Gibbs sampler for the 1-D Ising model by geometric bound

“…Frigessi et al (1992) showed that the second largest eigenvalue, β 1 , corresponds to minimizing the worst-case asymptotic variance for reversible MCMC algorithms. Many reports used the asymptotic variance to study the convergence rate of the MCMC algorithms (Diaconis and Stroock, 1991;Sinclair, 1992;Frigessi et al, 1993;Ingrassia, 1994;Mengersen and Tweedie, 1996;Greenwood et al, 1998;Roberts and Rosenthal, 2008;Chen et al, 2012;Chen and Hwang, 2013).…”

On the Convergence Time of Some Non-Reversible Markov Chain Monte Carlo Methods

“…Initialize in X 0 ∼ µ 0 and let X t = x Propose Y ∼ Q(x, · ) y Set X t+1 = y with probability A(x, y) = 1 ∧ R(x, y) where R(x, y) := π(y)Q(y, x) π(x)Q(x, y) if π(x)Q(x, y) = 0 1 otherwise (4) If the proposal is rejected, set X t+1 = x Nevertheless, as the DBC (2) imposes that the joint probabilities Pr(X t ∈ A, X t+1 ∈ B) and Pr(X t ∈ B, X t+1 ∈ A) are equal, reversibility may prevent the Markov chain from roaming efficiently through the state space, especially when π's topology is irregular. This is illustrated by the following example.…”

“…Recent works have shown that the asymptotic efficiency of MCMC algorithms using a non-reversible Markov chain is typically lower than those using reversible dynamic (see for instance [11,4,5,10]). Several methods have been developed to construct such chains (see [1,3,8,18], amongst others).…”