Monotone Nonlinear Probabilistic Models, with Credit Applications

We establish novel necessary and sufficient conditions for a particularly tractable concept of monotonic dependence of an exponential conditional probability model on its explanatory variables. We introduce convex programming methods to calibrate such models (which are monotone in conditional mean, conditional median and conditional mode). Armed with such methods, the model builder can incorporate prior knowledge, with respect to monotone dependence on a subset of the explanatory variables, into a broad class of nonlinear conditional probability models (including two-state models, as well as models with more than two states). Our method amounts to a novel form of regularization that, as our benchmarking experiments in the credit modeling context show, allows us to calibrate models that perform better than linear models, while maintaining the monotonic behavior that many users find reassuring