Convergence Properties of Perturbed Markov Chains

1998 ◽  
Vol 35 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Gareth O. Roberts ◽  
Jeffrey S. Rosenthal ◽  
Peter O. Schwartz
Keyword(s):  
Computer Simulation ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Rates Of Convergence ◽  
Geometric Ergodicity ◽  
Roundoff Error ◽  
Convergence Properties ◽  
Small Perturbations ◽  
Monte Carlo Algorithms

In this paper, we consider the question of which convergence properties of Markov chains are preserved under small perturbations. Properties considered include geometric ergodicity and rates of convergence. Perturbations considered include roundoff error from computer simulation. We are motivated primarily by interest in Markov chain Monte Carlo algorithms.

Convergence Properties of Perturbed Markov Chains

1998 ◽  
Vol 35 (01) ◽  
pp. 1-11 ◽  
Author(s):  
Gareth O. Roberts ◽  
Jeffrey S. Rosenthal ◽  
Peter O. Schwartz
Keyword(s):  
Computer Simulation ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Rates Of Convergence ◽  
Geometric Ergodicity ◽  
Roundoff Error ◽  
Convergence Properties ◽  
Small Perturbations ◽  
Monte Carlo Algorithms

In this paper, we consider the question of which convergence properties of Markov chains are preserved under small perturbations. Properties considered include geometric ergodicity and rates of convergence. Perturbations considered include roundoff error from computer simulation. We are motivated primarily by interest in Markov chain Monte Carlo algorithms.

Exact Markov Chain Monte Carlo Algorithms and Their Applications in Probabilistic Data Analysis and Inference

2009 ◽  
pp. 161-183
Author(s):  
Dominic Savio Lee
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Data Analysis ◽  
Markov Chain Monte Carlo ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Probabilistic Data ◽  
Perfect Sampling ◽  
Monte Carlo Algorithms ◽  
Sampling Algorithms

This chapter describes algorithms that use Markov chains for generating exact sample values from complex distributions, and discusses their use in probabilistic data analysis and inference. Its purpose is to disseminate these ideas more widely so that their use will become more widespread, thereby improving Monte Carlo simulation results and stimulating greater research interest in the algorithms themselves. The chapter begins by introducing Markov chain Monte Carlo (MCMC), which stems from the idea that sample values from a desired distribution f can be obtained from the stationary states of an ergodic Markov chain whose stationary distribution is f. To get sample values that have distribution f exactly, it is necessary to detect when the Markov chain has reached its stationary distribution. Under certain conditions, this can be achieved by means of coupled Markov chains—these conditions and the resulting exact MCMC or perfect sampling algorithms and their applications are described.

scholarly journals A Bayesian model for binary Markov chains

2004 ◽  
Vol 2004 (8) ◽  
pp. 421-429 ◽  
Author(s):  
Souad Assoudou ◽  
Belkheir Essebbar
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Bayesian Estimation ◽  
Bayesian Model ◽  
Transition Probabilities ◽  
Simulated Data ◽  
Bayesian Estimator ◽  
Jeffreys Prior ◽  
Data Set

This note is concerned with Bayesian estimation of the transition probabilities of a binary Markov chain observed from heterogeneous individuals. The model is founded on the Jeffreys' prior which allows for transition probabilities to be correlated. The Bayesian estimator is approximated by means of Monte Carlo Markov chain (MCMC) techniques. The performance of the Bayesian estimates is illustrated by analyzing a small simulated data set.

scholarly journals Using systematic sampling selection for Monte Carlo solutions of Feynman-Kac equations

2008 ◽  
Vol 40 (2) ◽  
pp. 454-472 ◽  
Author(s):  
Ivan Gentil ◽  
Bruno Rémillard
Keyword(s):  
Monte Carlo ◽  
Markov Chains ◽  
Selection Method ◽  
Systematic Sampling ◽  
Selection Methods ◽  
Convergence Properties ◽  
Main Motivation ◽  
Empirical Basis ◽  
Convergence Results ◽  
Selection For

While the convergence properties of many sampling selection methods can be proven, there is one particular sampling selection method introduced in Baker (1987), closely related to ‘systematic sampling’ in statistics, that has been exclusively treated on an empirical basis. The main motivation of the paper is to start to study formally its convergence properties, since in practice it is by far the fastest selection method available. We will show that convergence results for the systematic sampling selection method are related to properties of peculiar Markov chains.

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