Rates of convergence of the Hastings and Metropolis algorithms

By Kerrie L. Mengersen, Richard L. Tweedie. 1996

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Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families (2015, background): “Following [22, 30], it is sufﬁcient to show ˆ for some s > 0 , where µ ( · ) is a standard Gaussian measure.”
Variational MCMC (2001, methodology): “In fa t, we havethe following resultProposition 1 (Mengersen and Tweedie 1996, Theorem 2.1) The independent MHalgorithm onverges at a uniformly (geometri ) rate if there exists a onstant > 0 su hthat p( jxv;xv )bp( jxv;xv ) ; 2 supp(p( ))in whi h ase, kK(i)( ; :) pkTV 2 1 1 iwhere k kTV denotes the…”
Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (2012, background): “It is now well known that the IMH is uniformly (and geometrically) ergodic if and only if π(dx)/q(dx) is bounded [23].”
Fixed-Width Output Analysis for Markov Chain Monte Carlo (2006, background): “…whereas Christensen, Moller, and Waagepetersen (2001), Douc, Fort, Moulines, and Soulier (2004), Fort and Moulines (2000, 2003), Geyer (1999), Jarner and Hansen (2000), Jarner and Roberts (2002), Meyn and Tweedie (1994), and Mengersen and Tweedie (1996) analyzed Metropolis– Hastings algorithms.”
On the convergence of Hamiltonian Monte Carlo (2017, unknown): “The classical approach to derive irreducibility of Hastings-Metropolis algorithms on Rd, outlined for example in [18] [24], is to use that the proposal distribution admits a (sufficiently regular) transition density with respect to the Lebesgue measure.”
Statistical Modeling and Conceptualization of Visual Patterns (2003, methodology): “It was shown, through simple case, that the better the proposal probability approximates the posterior, the faster the algorithm converges [58].”
Convergence of adaptive mixtures of importance sampling schemes (2007, methodology): “MCMC algorithms somehow attenuate this difficulty by using local proposals like random walk kernels, but two drawbacks of those proposals are that they may take a long time to converge (Mengersen and Tweedie, 1996) and that their efficiency ultimately depend on the scale of the local exploration. Th”
Improved Sampling‐Importance Resampling and Reduced Bias Importance Sampling (2003, background): “Theorem 3 (Mengersen & Tweedie, 1996) If supx h(x) 1, then the probability density of MH with iid proposals drawn from q satisfies pnþ1 p TV 1 1 supx hðxÞ pn pk kTV: Board of the Foundation of the Scandinavian Journal of Statistics 2003.”
Inference in Hidden Markov Models (2006, methodology): “As reported in Mengersen and Tweedie (1996), the performance of an independent Metropolis-Hastings algorithm will vary widely, depending on, in particular, whether or not the importance ratio π(ξ)/rind(ξ) is bounded (which is also the condition required for applying the accept-reject algorithm).”
Quantitative bounds on convergence of time-inhomogeneous Markov chains (2004, methodology): “The RWMH algorithm on R was extensively studied by Mengersen and Tweedie (1996), who showed that the transition kernelsKγ are πγ -irreducible (Lemma 1.”
Particle Markov chain Monte Carlo methods (2010, methodology): “For an IMH algorithm this implies uniform geometric ergodicity towards π̃ , with a rate at least 1−Z=Πn=1Cn; see for example Mengersen and Tweedie (1996), theorem 2.”
Sensitivity and convergence of uniformly ergodic Markov chains (2005, methodology): “2) is obtained through minorization conditions [19, Chapter 16] (bounds of this type were used in [16] and [30]).”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, methodology): “Based a result by Roberts and Tweedie (1996b), also used in Mengersen and Tweedie (1996), namely that an MCMC chain cannot be geometrically ergodic when there exist almost-absorbing states, they derive that (under some technical assumptions) the ﬁrst two versions above cannot be variance-bounding…”
Honest Exploration of Intractable Probability Distributions via Markov Chain Monte Carlo (2001, background): “Mengersen and Tweedie (1996) show that if there exists a K > 0 such that”
Examples of Adaptive MCMC (2009, background): “A result of Mengersen and Tweedie (1996) says, essentially, that a random-walk Met-ropolis (RWM) algorithm for a density π will be geometrically ergodic if and only if π has exponential or sub-exponential tails, i.e. satisﬁes log π ( x ) − log π ( y ) ≥ η ( y − x ) , y > x ≥ x 1 (5) for some x 1 >…”