Fall Seminar Series - November 18, 2004
University of Minnesota
School of Statistics
College of Liberal Arts
Improved
Minimax Prediction Under Kullback-Leibler Loss
Edward George
Department of Statistics
University of Pennsylvania
Thursday, November 18, 2004
3:30 PM, 115
Ford Hall
Minneapolis, East Bank Campus
Social at 3:00 PM, 300 Ford Hall
Abstract
Let
$X | \mu \sim N_p(\mu, v_x I)$ and $Y | \mu \sim N_p(\mu, v_y I)$ be
independent $p$-dimensional multivariate normal vectors with common
unknown
mean
$\mu$, and let $p(x | \mu)$ and $p(y | \mu)$ denote the conditional
densities of
$X$ and $Y$. Based on only observing $X = x$, we consider the problem
of
obtaining a predictive distribution $\hat p(y | x)$ for $Y$ that is
close
to
$p(y | \mu)$ as measured by Kullback-Leibler loss. The natural straw
man
for
this problem is the best invariant predictive distribution, the Bayes
rule
$p_U(y | x)$ under the uniform prior $\pi_U(\mu) \equiv 1$, which is
seen
to be
minimax. We show that $p_U(y | x)$ is dominated by any Bayes rules for
which the
square root of the marginal distribution is superharmonic. This yields
wide
classes of dominating predictive distributions including Bayes rules
under
superharmonic priors. These dominating predictive shrinkage
distributions
can be
constructed to adaptively shrink $p_U(y | x)$ towards arbitrary points
or
subspaces. Those procedures corresponding to superharmonic priors can
be
further
combined to obtain minimax multiple shrinkage predictive distributions
that
adaptively shrink $p_U(y | x)$ towards an arbitrary number of points or
subspaces. Fundamental similarities and differences with the parallel
theory of
estimating a multivariate normal mean under quadratic loss are
described
throughout. (This is joint work with Feng Liang, Duke U and and Xinyi
Xu, U
of
Penn).