Views
3 years ago

Monte Carlo integration with Markov chain - Department of Statistics

Monte Carlo integration with Markov chain - Department of Statistics

Monte Carlo integration with Markov chain - Department of

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

Introduction to Bayesian Data Analysis and Markov Chain Monte Carlo
Asteroid orbital inversion using Markov-chain Monte Carlo methods
To Do Motivation Monte Carlo Path Tracing Monte Carlo Path ...
An Introduction to Monte Carlo Methods in Statistical Physics.
Path Integral Monte Carlo approach to ultracold atomic gases
Bayesian Analysis with Monte Carlo Markov-Chain Methods
EFFICIENT RISK MANAGEMENT IN MONTE CARLO - Luca Capriotti
Past and Future of Monte Carlo in Medical Physics - Department of ...
Monte Carlo simulation inspired by computational ... - mcqmc 2012
How and why the Monte Carlo method works (pdf)
Monte Carlo Simulations: Efficiency Improvement Techniques and ...
Monte Carlo simulations for brachytherapy - Carleton University
Download Monte Carlo Oil And Gas Reserve Estimation - Lumenaut
Escalation Estimating Using Indices and Monte Carlo Simulation
Markov Chain Monte Carlo Methods for Statistical Inference
IRREDUCIBLE MARKOV CHAIN MONTE CARLO ... - LSE Statistics
Markov Chain Monte Carlo - Penn State Department of Statistics ...
Markov chain Monte Carlo methods - the IMM department
Markov Chain Monte Carlo for Statistical Inference - Materials ...
Markov Chains and Monte Carlo Methods - users-deprecated.aims ...
Tutorial on Markov Chain Monte Carlo Simulations and Their ...
Markov Chain Monte Carlo in Conditionally Gaussian State Space ...
Introduction to Markov Chain Monte Carlo & Gibbs Sampling
Markov chain Monte Carlo algorithms for Gaussian processes
Markov Chain Monte Carlo Lecture Notes
MCMCpack: Markov Chain Monte Carlo in R - Journal of Statistical ...
Markov Chain Monte Carlo and mixing rates - Department of ...
Introduction to Markov Chain Monte Carlo, with R