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Optimistic Distributionally Robust Optimization for Nonparametric Likelihood Approximation

Viet Anh Nguyen(viet-anh.nguyen***at***epfl.ch)
Shafieezadeh-Abadeh Soroosh(soroosh.shafiee***at***epfl.ch)
Man-Chung Yue(manchung.yue***at***polyu.edu.hk)
Daniel Kuhn(daniel.kuhn***at***epfl.ch)
Wolfram Wiesemann(ww***at***imperial.ac.uk)

Abstract: 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.

Keywords: optimistic likelihood, nonparametric Bayesian statistics, Wasserstein distance, Kullback-Leibler divergence, moment ambiguity set

Category 1: Stochastic Programming


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Entry Submitted: 10/23/2019
Entry Accepted: 10/23/2019
Entry Last Modified: 10/23/2019

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