Density Estimation in Infinite Dimensional Exponential Families
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In this paper, we consider an infinite dimensional exponential family, P of probability densities, which are parametrized by functions in a reproducing kernel Hilbert space, H and show it to be quite rich in the sense that a broad class of densities on R^d can be approximated arbitrarily well in Kullback-Leibler (KL) divergence by elements in P. The main goal of the paper is to estimate an unknown density, p_0 through an element in P. Standard techniques like maximum likelihood estimation (MLE) or pseudo MLE (based on the method of sieves), which are based on minimizing the KL divergence between p_0 and P, do not yield practically useful estimators because of their inability to efficiently handle the log-partition function. Instead, we propose an estimator, p̂_n based on minimizing the Fisher divergence, J(p_0 p) between p_0 and p∈P, which involves solving a simple finite-dimensional linear system. When p_0∈P, we show that the proposed estimator is consistent, and provide a convergence rate of n^-{2/3,2β+1/2β+2} in Fisher divergence under the smoothness assumption that p_0∈R(C^β) for some β> 0, where C is a certain Hilbert-Schmidt operator on H and R(C^β) denotes the image of C^β. We also investigate the misspecified case of p_0∉P and show that J(p_0p̂_n)→_p∈PJ(p_0 p) as n→∞, and provide a rate for this convergence under a similar smoothness condition as above. Through numerical simulations we demonstrate that the proposed estimator outperforms the non-parametric kernel density estimator, and that the advantage with the proposed estimator grows as d increases.
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