  This paper proposes a novel method to estimate individualised treatment
assignment rules. The method is designed to find rules that are stochastic,
reflecting uncertainty in estimation of an assignment rule and about its
welfare performance. Our approach is to form a prior distribution over
assignment rules, not over data generating processes, and to update this prior
based upon an empirical welfare criterion, not likelihood. The social planner
then assigns treatment by drawing a policy from the resulting posterior. We
show analytically a welfare-optimal way of updating the prior using empirical
welfare; this posterior is not feasible to compute, so we propose a variational
Bayes approximation for the optimal posterior. We characterise the welfare
regret convergence of the assignment rule based upon this variational Bayes
approximation, showing that it converges to zero at a rate of ln(n)/sqrt(n). We
apply our methods to experimental data from the Job Training Partnership Act
Study to illustrate the implementation of our methods.
