Learning Probabilistic models: An expected utility maximization approach

We consider the problem of learning a probabilistic model from the viewpoint of an expected utility maximizing decision maker/investor who would use the model to make decisions (bets), which result in well defined payoffs. In our new approach, we seek good out-of-sample model performance by considering a one-parameter family of Pareto optimal models, which we define in terms of consistency with the training data and consistency with a prior (benchmark) model. We measure the former by means of the large-sample distribution of a vector of sample-averaged features, and the latter by means of a generalized relative entropy. We express each Pareto optimal model as the solution of a strictly convex optimization problem and its strictly concave (and tractable) dual. Each dual problem is a regularized maximization of expected utility over a well-defined family of functions. Each Pareto optimal model is robust: maximizing worst-case outperformance relative to the benchmark model. Finally, we select the Pareto optimal model with maximum (out-of-sample) expected utility. We show that our method reduces to the minimum relative entropy method if and only if the utility function is a member of a three-parameter logarithmic family.