Estimating Density Models with Complex Truncation Boundaries

Truncated densities are probability density functions defined on truncated input domains. These densities share the same parametric form with their non-truncated counterparts up to a normalization term. However, normalization terms usually cannot be obtained in closed form for these distributions, due to complicated truncation domains. Score Matching is a powerful tool for fitting parameters in unnormalized models. However, it cannot be straightforwardly applied here as boundary conditions used to derive a tractable objective are usually not satisfied by truncated distributions. In this paper, we propose a maximally weighted Score Matching objective function which takes the geometry of the truncation boundary into account when fitting unnormalized density models. We show the weighting function that maximizes the objective function can be constructed easily and the boundary conditions for deriving a tradable objective are satisfied. Experiments on toy datasets and Chicago crime dataset show promising results.