Consider an object with arbitrary mass density. Its gravitational field can be computed exactly and uniquely via Newton's equations. Now, suppose measurements are taken on the gravitational field. Can we infer the mass density of the object? This problem is generally known as the inverse problem, and arises in a wide range of fields. In this talk, a statistical approach is taken to solving that problem. Specifically, instead of inferring mass density itself, the problem is cast into a classification framework, and it is shown that a number of linear and nonlinear classifiers can be employed to provide a conditional probability of class membership. The performance of the classifiers is gauged in a framework designed to assess the quality of the probabilities.