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Efficient Adaptive MCMC Through Precision Estimation

Journal article
Authors J. Wallin
David Bolin
Published in Journal of Computational and Graphical Statistics
Volume 27
Issue 4
Pages 887-897
ISSN 1061-8600
Publication year 2018
Published at Department of Mathematical Sciences
Pages 887-897
Language en
Links dx.doi.org/10.1080/10618600.2018.14...
Keywords AMCMC, Cholesky estimation, MALA, MCMC, MHRW, Online estimation, Partial correlation, covariance estimation, langevin, selection, models
Subject categories Probability Theory and Statistics

Abstract

The performance of Markov chain Monte Carlo (MCMC) algorithms like the Metropolis Hastings Random Walk (MHRW) is highly dependent on the choice of scaling matrix for the proposal distributions. A popular choice of scaling matrix in adaptive MCMC methods is to use the empirical covariance matrix (ECM) of previous samples. However, this choice is problematic if the dimension of the target distribution is large, since the ECM then converges slowly and is computationally expensive to use. We propose two algorithms to improve convergence and decrease computational cost of adaptive MCMC methods in cases when the precision (inverse covariance) matrix of the target density can be well-approximated by a sparse matrix. The first is an algorithm for online estimation of the Cholesky factor of a sparse precision matrix. The second estimates the sparsity structure of the precision matrix. Combining the two algorithms allows us to construct precision-based adaptive MCMC algorithms that can be used as black-box methods for densities with unknown dependency structures. We construct precision-based versions of the adaptive MHRW and the adaptive Metropolis adjusted Langevin algorithm and demonstrate the performance of the methods in two examples. Supplementary materials for this article are available online.

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