  In inference problems involving a multi-dimensional parameter $\theta$, it is
often natural to consider decision rules that have a risk which is invariant
under some group $G$ of permutations of $\theta$. We show that this implies
that the Bayes risk of the rule is {\em as if} the prior distribution of the
parameter is partially exchangeable with respect to $G$. We provide a
symmetrization technique for incorporating partial exchangeability of $\theta$
into a statistical model, without assuming any other prior information. We
refer to this technique as {\em shuffling}. Shuffling can be viewed as an
instance of empirical Bayes, where we estimate the (unordered) multiset of
parameter values $\{\theta_1,\theta_2,\dots,\theta_p\}$ while using a uniform
prior on $G$ for their ordering. Estimation of the multiset is a missing data
problem which can be tackled with a stochastic EM algorithm. We show that in
the special case of estimating the mean-value parameter in a regular
exponential family model, shuffling leads to an estimator that is a weighted
average of permuted versions of the usual maximum likelihood estimator. This is
a novel form of shrinkage.
