3,613 research outputs found
Control Variates for Reversible MCMC Samplers
A general methodology is introduced for the construction and effective
application of control variates to estimation problems involving data from
reversible MCMC samplers. We propose the use of a specific class of functions
as control variates, and we introduce a new, consistent estimator for the
values of the coefficients of the optimal linear combination of these
functions. The form and proposed construction of the control variates is
derived from our solution of the Poisson equation associated with a specific
MCMC scenario. The new estimator, which can be applied to the same MCMC sample,
is derived from a novel, finite-dimensional, explicit representation for the
optimal coefficients. The resulting variance-reduction methodology is primarily
applicable when the simulated data are generated by a conjugate random-scan
Gibbs sampler. MCMC examples of Bayesian inference problems demonstrate that
the corresponding reduction in the estimation variance is significant, and that
in some cases it can be quite dramatic. Extensions of this methodology in
several directions are given, including certain families of Metropolis-Hastings
samplers and hybrid Metropolis-within-Gibbs algorithms. Corresponding
simulation examples are presented illustrating the utility of the proposed
methods. All methodological and asymptotic arguments are rigorously justified
under easily verifiable and essentially minimal conditions.Comment: 44 pages; 6 figures; 5 table
Coupling Control Variates for Markov Chain Monte Carlo
We show that Markov couplings can be used to improve the accuracy of Markov
chain Monte Carlo calculations in some situations where the steady-state
probability distribution is not explicitly known. The technique generalizes the
notion of control variates from classical Monte Carlo integration. We
illustrate it using two models of nonequilibrium transport
Regression estimators in simulation
sampling;simulation;queuing systems;operations research
Large deviation asymptotics and control variates for simulating large functions
Consider the normalized partial sums of a real-valued function of a
Markov chain, The
chain takes values in a general state space ,
with transition kernel , and it is assumed that the Lyapunov drift condition
holds: where , , the set is small and dominates . Under these
assumptions, the following conclusions are obtained: 1. It is known that this
drift condition is equivalent to the existence of a unique invariant
distribution satisfying , and the law of large numbers
holds for any function dominated by :
2. The lower error
probability defined by , for , ,
satisfies a large deviation limit theorem when the function satisfies a
monotonicity condition. Under additional minor conditions an exact large
deviations expansion is obtained. 3. If is near-monotone, then
control-variates are constructed based on the Lyapunov function , providing
a pair of estimators that together satisfy nontrivial large asymptotics for the
lower and upper error probabilities. In an application to simulation of queues
it is shown that exact large deviation asymptotics are possible even when the
estimator does not satisfy a central limit theorem.Comment: Published at http://dx.doi.org/10.1214/105051605000000737 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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