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# Maximum Likelihood with Coarse Data based on Robust Optimisation

*Proceedings of the Tenth International Symposium on Imprecise Probability: Theories and Applications*, PMLR 62:169-180, 2017.

#### Abstract

This paper deals with the problem of probability estimation in the context of coarse data. Probabilities are estimated using the maximum likelihood principle. Our approach presupposes that each imprecise observation underlies a precise one, and that the uncertainty that pervades its observation is epistemic, rather than representing noise. As a consequence, the likelihood function of the ill-observed sample is set-valued. In this paper, we apply a robust optimization method to find a safe plausible estimate of the probabilities of elementary events on finite state spaces. More precisely we use a maximin criterion on the imprecise likelihood function. We show that there is a close connection between the robust maximum likelihood strategy and the maximization of entropy among empirical distributions compatible with the incomplete data. A mathematical model in terms of maximal flow on graphs, based on duality theory, is proposed. It results in a linear objective function and convex constraints. This result is somewhat surprizing since maximum entropy problems are known to be complex due to the maximization of a concave function on a convex set.