Marginal likelihoods for finite-support Huber contamination

Abstract

For Huber contamination on a known finite sample space, the unrestricted contaminating law is a probability vector on the support atoms, and domination over all measurable subsets reduces to atomwise inequalities. Placing a Dirichlet prior on this probability vector and a Beta prior on the contamination proportion gives an exact marginal likelihood for the structural parameter after analytic integration of both nuisance quantities. The likelihood is a finite weighted sum over allocations of the observed counts between the structural and contaminating components. For fixed support size, this sum and its score can be evaluated by a dynamic program with quadratic cost in the sample size, enabling gradient-based posterior sampling.