General state space Markov chains and MCMC algorithms

By Gareth O. Roberts, Jeffrey S. Rosenthal. 2004

913 citations recorded by the source corpus.

This paper surveys various results about Markov chains on gen- eral (non-countable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sucient conditions for geomet- ric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift con- ditions. Necessary and sucient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equa- tion or direct regeneration constructions. Finally, optimal scaling and weak convergence results for Metropolis-Hastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/be55a1132d4db91fb550383ea8246de1898fe761

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

A Practical Sequential Stopping Rule for High-Dimensional Markov Chain Monte Carlo (2016, background): “Under certain regularity conditions (Chan and Geyer 1994; Tierney 1994; Jones 2004; Roberts and Rosenthal 2004), we can obtain a marginal Markov chain central limit theorem (CLT) for the sampling distribution of an unknown MCSE.”
Control variates for estimation based on reversible Markov chain Monte Carlo samplers (2012, background): “The chain {Xn} is ψ irreducible and aperiodic, with unique invariant measure π, and there are functions V : X → [0, ∞/, W : X → [1, ∞/, a small set C ∈B and a finite constant b> 0, such that condition .36/ holds and π.V 2/<∞: .37/ Although these conditions may seem somewhat involved, their verificat”
Stochastic Gradient Markov Chain Monte Carlo (2019, background): “Under mild conditions, it is easy to show that asymptotically the samples generated from the Markov chain converge to the posterior distribution (Roberts et al., 2004) and for many of the popular MCMC algorithms, rates of convergence based on geometric ergodicity have been established (see Meyn et a”
Inference in hidden Markov models (2010, background): “We give here a proof due to Rosenthal (1995), based on pathwise coupling (see Rosenthal, 2001; Roberts and Rosenthal, 2004).”
A Conceptual Introduction to Hamiltonian Monte Carlo (2017, background): “Formally, the sufficient condition that guarantees the idealized behavior, most importantly a Central Limit Theorem for the Markov chain Monte Carlo estimators, is known as geometric ergodicity (Roberts and Rosenthal, 2004).”
Fixed-Width Output Analysis for Markov Chain Monte Carlo (2006, background): “The MCMC community has expended considerable effort in establishing such mixing conditions for a variety of samplers (see Jones and Hobert 2001; Roberts and Rosenthal 1998, 2004 for some references and discussion).”
Adaptive Gibbs samplers and related MCMC methods (2011, methodology): “Indeed, (i) clearly holds since α n → α := ( , ), and (ii) follows immediately from the standard Markov chain properties of irreducibility and aperiodicity (c.f. [33, 43]).”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, methodology): “First, standard measures of efficiency of Markovian kernels, like the total variation convergence rate (c.f. Meyn and Tweedie 2009; Roberts and Rosenthal 2004), L2(π) spectral gap (Diaconis and Stroock 1991; Roberts 1996; Saloff-Coste 1997; Levin et al.”
Optimal Proposal Distributions and Adaptive MCMC (2008, methodology): “Many applications of MCMC make use of such Markov chain limit theorems as the Weak Law of Large Numbers (WLLN), Strong Law of Large Numbers (SLLN), and Central Limit Theorem (CLT), in order to guarantee good asymptotic estimates and estimate standard errors (see e.g. Tierney, 1994; Jones and Hobert,”
Quantitative bounds for Markov chain convergence: Wasserstein and total variation distances (2010, methodology): “Many tools have been developed for convergence in TV, involving probabilistic methods (for example, coupling, strong uniform times; see [5, 13, 19] for reviews), analytic methods (spectral analysis, Fourier analysis, operator theory; see [5, 21]) and geometric methods (path bounds, isoperimetry; see”
Convergence rates and asymptotic standard errors for Markov chain Monte Carlo algorithms for Bayesian probit regression (2007, methodology): “Together with the results of Roberts and Rosenthal (1997), this theorem implies that the AC algorithm has a CLT for every f ∈L2{π.β|y/}. To use this theory to calculate standard errors, we require a consistent estimator of the asymptotic variance, σ2. This topic will be addressed in Section 4. In th”
The Geometric Foundations of Hamiltonian Monte Carlo (2014, background): “, qN), of the Markov chain yields Markov chain Monte Carlo estimators of any expectation [50,61].”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, methodology): “More precisely, for distributions with suﬃciently smooth contours, MALA is geometrically ergodic (c.f. Roberts and Rosenthal (2004)) if the tails of π decay as exp {−| θ | β } with β ∈ [1 , 2], while the random walk Metropolis al-gorithm is geometrically ergodic for all β ≥ 1 (Roberts and Tweedie…”