18,663 research outputs found
Batch means and spectral variance estimators in Markov chain Monte Carlo
Calculating a Monte Carlo standard error (MCSE) is an important step in the
statistical analysis of the simulation output obtained from a Markov chain
Monte Carlo experiment. An MCSE is usually based on an estimate of the variance
of the asymptotic normal distribution. We consider spectral and batch means
methods for estimating this variance. In particular, we establish conditions
which guarantee that these estimators are strongly consistent as the simulation
effort increases. In addition, for the batch means and overlapping batch means
methods we establish conditions ensuring consistency in the mean-square sense
which in turn allows us to calculate the optimal batch size up to a constant of
proportionality. Finally, we examine the empirical finite-sample properties of
spectral variance and batch means estimators and provide recommendations for
practitioners.Comment: Published in at http://dx.doi.org/10.1214/09-AOS735 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Sufficient burn-in for Gibbs samplers for a hierarchical random effects model
We consider Gibbs and block Gibbs samplers for a Bayesian hierarchical
version of the one-way random effects model. Drift and minorization conditions
are established for the underlying Markov chains. The drift and minorization
are used in conjunction with results from J. S. Rosenthal [J. Amer. Statist.
Assoc. 90 (1995) 558-566] and G. O. Roberts and R. L. Tweedie [Stochastic
Process. Appl. 80 (1999) 211-229] to construct analytical upper bounds on the
distance to stationarity. These lead to upper bounds on the amount of burn-in
that is required to get the chain within a prespecified (total variation)
distance of the stationary distribution. The results are illustrated with a
numerical example
Markov Chain Monte Carlo: Can We Trust the Third Significant Figure?
Current reporting of results based on Markov chain Monte Carlo computations
could be improved. In particular, a measure of the accuracy of the resulting
estimates is rarely reported. Thus we have little ability to objectively assess
the quality of the reported estimates. We address this issue in that we discuss
why Monte Carlo standard errors are important, how they can be easily
calculated in Markov chain Monte Carlo and how they can be used to decide when
to stop the simulation. We compare their use to a popular alternative in the
context of two examples.Comment: Published in at http://dx.doi.org/10.1214/08-STS257 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Evaluation of Formal posterior distributions via Markov chain arguments
We consider evaluation of proper posterior distributions obtained from
improper prior distributions. Our context is estimating a bounded function
of a parameter when the loss is quadratic. If the posterior mean of
is admissible for all bounded , the posterior is strongly
admissible. We give sufficient conditions for strong admissibility. These
conditions involve the recurrence of a Markov chain associated with the
estimation problem. We develop general sufficient conditions for recurrence of
general state space Markov chains that are also of independent interest. Our
main example concerns the -dimensional multivariate normal distribution with
mean vector when the prior distribution has the form on the parameter space . Conditions on for strong
admissibility of the posterior are provided.Comment: Published in at http://dx.doi.org/10.1214/07-AOS542 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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