A Financial Approach to Machine Learning with Applications to Credit Risk

We review a particular financially motivated method for evaluating probabilistic models and learning such models from data. We adopt the viewpoint of an expected-utility-maximizing investor who would use the model to make decisions (bets) that result in well-defined payoffs. In order to evaluate a particular model, we assume that there is an investor who believes the model. This investor allocates his assets so as to maximize his expected utility according to his beliefs, i.e., the investor allocates so as to maximize the expectation of his utility under the model probability measure. We then measure the success of the investor's investment strategy in terms of the average utility the strategy provides on an out-of-sample data set. For an investor with a utility function in a certain logarithmic family, the resulting performance measure is the likelihood ratio. In the learning approach that we review here, we consider 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, which is a regularized maximization of expected utility over a well-defined family of functions. Each Pareto optimal model is robust in the sense that it maximizes the worst-case outperformance relative to the benchmark model. We select the Pareto optimal model with maximum (out-of-sample) expected utility. We review the application of this learning method to two important credit risk problems: estimating conditional default probabilities, and estimating conditional probabilities for recovery rates of defaulted debt