3,962 research outputs found
Nonasymptotic bounds on the mean square error for MCMC estimates via renewal techniques
The Nummellinâs split chain construction allows to decompose a Markov
chain Monte Carlo (MCMC) trajectory into i.i.d. "excursions". Regenerative MCMC
algorithms based on this technique use a random number of samples. They have
been proposed as a promising alternative to usual fixed length simulation [25, 33,
14]. In this note we derive nonasymptotic bounds on the mean square error (MSE)
of regenerative MCMC estimates via techniques of renewal theory and sequential
statistics. These results are applied to costruct confidence intervals. We then focus
on two cases of particular interest: chains satisfying the Doeblin condition and a geometric
drift condition. Available explicit nonasymptotic results are compared for
different schemes of MCMC simulation
Relative fixed-width stopping rules for Markov chain Monte Carlo simulations
Markov chain Monte Carlo (MCMC) simulations are commonly employed for
estimating features of a target distribution, particularly for Bayesian
inference. A fundamental challenge is determining when these simulations should
stop. We consider a sequential stopping rule that terminates the simulation
when the width of a confidence interval is sufficiently small relative to the
size of the target parameter. Specifically, we propose relative magnitude and
relative standard deviation stopping rules in the context of MCMC. In each
setting, we develop sufficient conditions for asymptotic validity, that is
conditions to ensure the simulation will terminate with probability one and the
resulting confidence intervals will have the proper coverage probability. Our
results are applicable in a wide variety of MCMC estimation settings, such as
expectation, quantile, or simultaneous multivariate estimation. Finally, we
investigate the finite sample properties through a variety of examples and
provide some recommendations to practitioners.Comment: 24 page
The role of statistical methodology in simulation
statistical methods;simulation;operations research
Fixed-width output analysis for Markov chain Monte Carlo
Markov chain Monte Carlo is a method of producing a correlated sample in
order to estimate features of a target distribution via ergodic averages. A
fundamental question is when should sampling stop? That is, when are the
ergodic averages good estimates of the desired quantities? We consider a method
that stops the simulation when the width of a confidence interval based on an
ergodic average is less than a user-specified value. Hence calculating a Monte
Carlo standard error is a critical step in assessing the simulation output. We
consider the regenerative simulation and batch means methods of estimating the
variance of the asymptotic normal distribution. We give sufficient conditions
for the strong consistency of both methods and investigate their finite sample
properties in a variety of examples
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