Generalised Score Matching on Convex Domains

Abstract

Score matching avoids computing the normalising constant that maximum-likelihood estimation requires. On constrained domains, its generalised variants weight the Fisher divergence so that boundary terms vanish. We derive generalised score matching on open convex subsets of ${\mathbb{R}}^d$ as the small-neighbourhood limit of minimum probability flow, in which the geometry of the neighbourhoods determines the weight. Every $C^2$ positive definite weight arises in this way, including those of classical score matching on ${\mathbb{R}}^d$ and of its variants for non-negative data on ${\mathbb{R}}_{+}^d$. For exponential families, we extend the standard convexity, consistency and asymptotic normality results to every such weight and show that the estimator converges to the true parameter under certain boundary conditions. For a truncated Gaussian on a polytope and a Dirichlet distribution on the simplex, proposed estimators attain the lowest median error of all methods compared, in at least 42 of 50 ground-truth configurations.