Optimistic Distributionally Robust Optimization for Nonparametric Likelihood Approximation

The likelihood function is a fundamental component in Bayesian statistics. However, evaluating the likelihood of an observation is computationally intractable in many applications. In this paper, we propose a non-parametric approximation of the likelihood that identifies a probability measure which lies in the neighborhood of the nominal measure and that maximizes the probability of observing the given sample point. We show that when the neighborhood is constructed by the Kullback-Leibler divergence, by moment conditions or by the Wasserstein distance, then our optimistic likelihood can be determined through the solution of a convex optimization problem, and it admits an analytical expression in particular cases. We also show that the posterior inference problem with our optimistic likelihood approximation enjoys strong theoretical performance guarantees, and it performs competitively in a probabilistic classification task.

Published in:
Advances In Neural Information Processing Systems 32 (Nips 2019), 32
Presented at:
33rd Conference on Neural Information Processing Systems (NeurIPS), Vancouver, Canada, December 8-14, 2019

 Record created 2019-09-03, last modified 2020-07-10

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