Convergence of adaptive and interacting Markov chain Monte Carlo algorithms

Abstract

Adaptive and interacting Markov chain Monte Carlo algorithms (MCMC) have been recently introduced in the literature. These novel simulation algorithms are designed to increase the simulation efficiency to sample complex distributions. Motivated by some recently introduced algorithms (such as the adaptive Metropolis algorithm and the interacting tempering algorithm), we develop a general methodological and theoretical framework to establish both the convergence of the marginal distribution and a strong law of large numbers. This framework weakens the conditions introduced in the pioneering paper by Roberts and Rosenthal [J. Appl. Probab. 44 (2007) 458--475]. It also covers the case when the target distribution $\pi$ is sampled by using Markov transition kernels with a stationary distribution that differs from $\pi$.

Published in at http://dx.doi.org/10.1214/11-AOS938 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (10)

• General state space Markov chains and MCMC algorithms
• On the ergodicity properties of some adaptive MCMC algorithms
• Renewal theory and computable convergence rates for geometrically ergodic Markov chains
• Equi-energy sampler with applications in statistical inference and statistical mechanics
• Quantitative bounds on convergence of time-inhomogeneous Markov chains
• Convergence of adaptive and interacting Markov chain Monte Carlo algorithms
• On the ergodicity of the adaptive Metropolis algorithm on unbounded domains
• Sequentially interacting Markov chain Monte Carlo methods
• On nonlinear Markov chain Monte Carlo
• A cautionary tale on the efficiency of some adaptive Monte Carlo schemes

Cited by in corpus (26)

• Convergence of adaptive and interacting Markov chain Monte Carlo algorithms
• Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations
• On perturbed proximal gradient algorithms
• Non-asymptotic Analysis of Biased Stochastic Approximation Scheme
• Minimising MCMC variance via diffusion limits, with an application to simulated tempering
• Computation for Latent Variable Model Estimation: A Unified Stochastic Proximal Framework
• Accelerated dimension-independent adaptive Metropolis
• Adaptive parallel tempering algorithm
• The Wang-Landau Algorithm as Stochastic Optimization and Its Acceleration
• A Contour Stochastic Gradient Langevin Dynamics Algorithm for Simulations of Multi-modal Distributions
• Navigating to the Best Policy in Markov Decision Processes
• Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo
• Adaptive MCMC with online relabeling
• On the convergence of an improved and adaptive kinetic simulated annealing
• Configurational entropy, transition rates, and optimal interactions for rapid folding in coarse-grained model proteins
• Finite Sample Properties of Adaptive Markov Chains via Curvature
• On the Convergence of Consensus Algorithms with Markovian Noise and Gradient Bias
• In Search of Lost (Mixing) Time: Adaptive Markov chain Monte Carlo schemes for Bayesian variable selection with very large p
• Efficient preconditioned stochastic gradient descent for estimation in latent variable models
• Self-Healing Umbrella Sampling: Convergence and efficiency
• Adaptive Equi-Energy Sampler : Convergence and Illustration
• Adaptive Metropolis Algorithm Using Variational Bayesian Adaptive Kalman Filter
• On convergence of 1D Markov diffusions to heavy-tailed invariant density
• Adaptive Component-wise Multiple-Try Metropolis Sampling
• Adaptive Markov Chain Monte Carlo for Auxiliary Variable Method and Its Application to Parallel Tempering
• Forgetting the starting distribution in finite interacting tempering