new paper available

[David Haussler]

A new paper by D. Haussler and M. Opper entitled Mutual Information, 
Metric Entropy, and Risk in Estimation of Probability Distributions
is available on the web at
     
     http://www.cse.ucsc.edu/~sherrod/ml/research.html

An abstract is given below (for those who read LaTex) -David

___________________
Abstract:

$\{P_{Y|\theta}: \theta \in \Theta\}$ is a set of probability distributions
(with a common dominating measure)
on a complete separable metric space $Y$. A state $\theta^* \in \Theta$ is
chosen by Nature. A statistician gets $n$ independent observations
$Y_1, \ldots, Y_n$ distributed according to $P_{Y|\theta^*}$ and
produces an estimated distribution $\hat{P}$ for $P_{Y|\theta^*}$.
The statistician suffers a loss based on a measure of the distance between
the estimated distribution and the true distribution.
We examine the Bayes and minimax risk of this game for various loss functions,
including the relative entropy, the squared Hellinger distance, and the
$L_1$ distance. We also look at the cumulative
relative entropy risk over the distributions estimated during the first $n$
observations. Here the Bayes risk is the mutual information
between the random parameter $\Theta^*$ and the observations
$Y_1, \ldots, Y_n$.
New bounds on this mutual information are given in terms of the Laplace
transform of the Hellinger distance between $P_{Y|\theta}$ and
$P_{Y|\theta^*}$.  From these, bounds on the minimax risk are given
in terms of the metric entropy of $\Theta$
with respect to the Hellinger distance. The assumptions required
for these bounds are very general and do not depend
on the choice of the dominating measure. They apply to both finite and
infinite dimensional $\Theta$. They apply in some cases where $Y$ is
infinite dimensional, in some cases where $Y$ is not
compact, in some cases where the distributions are not smooth,
and in some parametric cases where asymptotic normality of the posterior
distribution fails.