# Ergodic averages for monotone functions using upper and lower dominating processes

Jesper Møller, Kerrie Mengersen

Research output: Contribution to journalJournal articleResearchpeer-review

### Abstract

We show how the mean of a monotone function (defined on a state space equipped with a partial ordering) can be estimated, using ergodic averages calculated from upper and lower dominating processes of a stationary irreducible Markov chain. In particular, we do not need to simulate the stationary Markov chain and we eliminate the problem of whether an appropriate burn-in is determined or not. Moreover, when a central limit theorem applies, we show how confidence intervals for the mean can be estimated by bounding the asymptotic variance of the ergodic average based on the equilibrium chain. Our methods are studied in detail for three models using Markov chain Monte Carlo methods and we also discuss various types of other models for which our methods apply.
Original language English Baysian Analysis 2 4 761-782 22 1936-0975 Published - 2007

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Ergodic Averages
Monotone Function
Markov processes
Markov chain
Burn-in
Partial ordering
Markov Chain Monte Carlo Methods
Asymptotic Variance
Central limit theorem
Confidence interval
State Space
Eliminate
Monte Carlo methods
Model

### Cite this

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title = "Ergodic averages for monotone functions using upper and lower dominating processes",
abstract = "We show how the mean of a monotone function (defined on a state space equipped with a partial ordering) can be estimated, using ergodic averages calculated from upper and lower dominating processes of a stationary irreducible Markov chain. In particular, we do not need to simulate the stationary Markov chain and we eliminate the problem of whether an appropriate burn-in is determined or not. Moreover, when a central limit theorem applies, we show how confidence intervals for the mean can be estimated by bounding the asymptotic variance of the ergodic average based on the equilibrium chain. Our methods are studied in detail for three models using Markov chain Monte Carlo methods and we also discuss various types of other models for which our methods apply.",
author = "Jesper M{\o}ller and Kerrie Mengersen",
year = "2007",
language = "English",
volume = "2",
pages = "761--782",
journal = "Bayesian Analysis",
issn = "1936-0975",
publisher = "Project Euclid",
number = "4",

}

In: Baysian Analysis, Vol. 2, No. 4, 2007, p. 761-782.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Ergodic averages for monotone functions using upper and lower dominating processes

AU - Møller, Jesper

AU - Mengersen, Kerrie

PY - 2007

Y1 - 2007

N2 - We show how the mean of a monotone function (defined on a state space equipped with a partial ordering) can be estimated, using ergodic averages calculated from upper and lower dominating processes of a stationary irreducible Markov chain. In particular, we do not need to simulate the stationary Markov chain and we eliminate the problem of whether an appropriate burn-in is determined or not. Moreover, when a central limit theorem applies, we show how confidence intervals for the mean can be estimated by bounding the asymptotic variance of the ergodic average based on the equilibrium chain. Our methods are studied in detail for three models using Markov chain Monte Carlo methods and we also discuss various types of other models for which our methods apply.

AB - We show how the mean of a monotone function (defined on a state space equipped with a partial ordering) can be estimated, using ergodic averages calculated from upper and lower dominating processes of a stationary irreducible Markov chain. In particular, we do not need to simulate the stationary Markov chain and we eliminate the problem of whether an appropriate burn-in is determined or not. Moreover, when a central limit theorem applies, we show how confidence intervals for the mean can be estimated by bounding the asymptotic variance of the ergodic average based on the equilibrium chain. Our methods are studied in detail for three models using Markov chain Monte Carlo methods and we also discuss various types of other models for which our methods apply.

M3 - Journal article

VL - 2

SP - 761

EP - 782

JO - Bayesian Analysis

JF - Bayesian Analysis

SN - 1936-0975

IS - 4

ER -