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Density Estimation in Infinite Dimensional Exponential Families

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In this work, we consider the problem of estimating densities in an infinite dimensional exponential family indexed by functions in a reproducing kernel Hilbert space. Since standard techniques like maximum likelihood estimation (MLE) or pseudo MLE (based on the method of sieves) do not yield practically useful estimators, we propose an estimator based on the minimization of Fisher divergence, which involves solving a simple finite dimensional linear system. We show that the proposed estimator is consistent, and provide convergence rates under smoothness assumptions (precisely, under the assumption that the true parameter or function that generates the data generating distribution lies in the image of a certain covariance operator). We also empirically demonstrate that the proposed method outperforms the standard non-parametric kernel density estimator. Joint work with Kenji Fukumizu, Arthur Gretton and Aapo Hyvarinen.

This talk is part of the Machine Learning @ CUED series.

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