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Monte Carlo integration with Markov chain - Department of Statistics

Journal **of** Statistical Planning and Inference 138 (2008) 1967 – 1980 www.elsevier.com/locate/jspi **Monte** **Carlo** **integration** **with** **Markov** **chain** Zhiqiang Tan **Department** **of** Biostatistics, Bloomberg School **of** Public Health, 615 North Wolfe Street, Johns Hopkins University, Baltimore, MD 21205, USA Received 20 April 2006; received in revised form 25 April 2007; accepted 18 July 2007 Available online 15 August 2007 Abstract There are two conceptually distinct tasks in **Markov** **chain** **Monte** **Carlo** (MCMC): a sampler is designed for simulating a **Markov** **chain** and then an estimator is constructed on the **Markov** **chain** for computing integrals and expectations. In this article, we aim to address the second task by extending the likelihood approach **of** Kong et al. for **Monte** **Carlo** **integration**. We consider a general **Markov** **chain** scheme and use partial likelihood for estimation. Basically, the **Markov** **chain** scheme is treated as a random design and a stratified estimator is defined for the baseline measure. Further, we propose useful techniques including subsampling, regulation, and amplification for achieving overall computational efficiency. Finally, we introduce approximate variance estimators for the point estimators. The method can yield substantially improved accuracy compared **with** Chib’s estimator and the crude **Monte** **Carlo** estimator, as illustrated **with** three examples. © 2007 Elsevier B.V. All rights reserved. Keywords: Gibbs sampling; Importance sampling; **Markov** **chain** **Monte** **Carlo**; Partial likelihood; Stratification; Variance estimation 1. Introduction **Markov** **chain** **Monte** **Carlo** (MCMC) has been extensively used in statistics and other scientific fields. A key idea is to simulate a **Markov** **chain** rather than a simple random sample for **Monte** **Carlo** **integration**. There are two conceptually distinct tasks: a sampler is designed for simulating a **Markov** **chain** converging to a target distribution and then an estimator is constructed on the **Markov** **chain** for computing integrals and expectations. The first task has been actively researched such as finding effective sampling algorithms and diagnosing convergence in various MCMC applications (e.g. Gilks et al., 1996; Liu, 2001). In this article, we shall be concerned **with** the second task, making efficient inference given a **Markov** **chain**. Suppose that q(x) is a nonnegative function on a state space X and its integral Z = ∫ q(x)dμ 0 is analytically intractable **with** respect to a baseline measure μ 0 . An MCMC algorithm can be applied to simulate a **Markov** **chain** (x 1 ,...,x n ) converging to the probability distribution **with** density p(x) = q(x) Z , E-mail address: ztan@jhsph.edu. 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2007.07.013

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