Information Bounds for Gibbs Samplers

If we wish to estimate efficiently the expectation of an arbitrary function on the basis of the output of a Gibbs sampler, which is better: deterministic or random sweep? In each case we calculate the asymptotic variance of the empirical estimator, the average of the function over the output, and de...

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Bibliographic Details
Published in:The Annals of statistics 1998-12, Vol.26 (6), p.2128-2156
Main Authors: Greenwood, Priscilla E., McKeague, Ian W., Wefelmeyer, Wolfgang
Format: Article
Language:English
Subjects:
62G20
62M05
Bayesian Analysis and Gibbs Sampling
Central limit theorem
Determinism
Efficient estimator
empirical estimator
Ergodic theory
Estimators
Exact sciences and technology
Inference from stochastic processes
time series analysis
Inner products
Markov chain Monte Carlo
Markov chains
Mathematics
Metropolitan areas
Nonparametric inference
Probability and statistics
Random sampling
Sciences and techniques of general use
Statistical variance
Statistics
Variance
variance bound
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Summary:If we wish to estimate efficiently the expectation of an arbitrary function on the basis of the output of a Gibbs sampler, which is better: deterministic or random sweep? In each case we calculate the asymptotic variance of the empirical estimator, the average of the function over the output, and determine the minimal asymptotic variance for estimators that use no information about the underlying distribution. The empirical estimator has noticeably smaller variance for deterministic sweep. The variance bound for random sweep is in general smaller than for deterministic sweep, but the two are equal if the target distribution is continuous. If the components of the target distribution are not strongly dependent, the empirical estimator is close to efficient under deterministic sweep, and its asymptotic variance approximately doubles under random sweep.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1024691464