2,398 research outputs found
Efficient sphere-covering and converse measure concentration via generalized coding theorems
Suppose A is a finite set equipped with a probability measure P and let M be
a ``mass'' function on A. We give a probabilistic characterization of the most
efficient way in which A^n can be almost-covered using spheres of a fixed
radius. An almost-covering is a subset C_n of A^n, such that the union of the
spheres centered at the points of C_n has probability close to one with respect
to the product measure P^n. An efficient covering is one with small mass
M^n(C_n); n is typically large. With different choices for M and the geometry
on A our results give various corollaries as special cases, including Shannon's
data compression theorem, a version of Stein's lemma (in hypothesis testing),
and a new converse to some measure concentration inequalities on discrete
spaces. Under mild conditions, we generalize our results to abstract spaces and
non-product measures.Comment: 29 pages. See also http://www.stat.purdue.edu/~yiannis
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
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