Information bounds for Gibbs samplers
P. E. Greenwood, I. W. McKeague, and W. Wefelmeyer
ABSTRACT
If we wish to efficiently estimate 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.