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Quantifying the difference between two probability density functions, $p$ and$q$, using available data, is a fundamental problem in Statistics and MachineLearning. A usual approach for addressing this problem is the likelihood-ratioestimation (LRE) between $p$ and $q$, which -- to our best knowledge -- hasbeen investigated mainly for the offline case. This paper contributes byintroducing a new framework for online non-parametric LRE (OLRE) for thesetting where pairs of iid observations $(x_t \sim p, x'_t \sim q)$ areobserved over time. The non-parametric nature of our approach has the advantageof being agnostic to the forms of $p$ and $q$. Moreover, we capitalize on therecent advances in Kernel Methods and functional minimization to develop anestimator that can be efficiently updated online. We provide theoreticalguarantees for the performance of the OLRE method along with empiricalvalidation in synthetic experiments.

Online non-parametric likelihood-ratio estimation by Pearson-divergence functional minimization