Density Estimation in Infinite Dimensional Exponential Families

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 ℝd 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, p0 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 p0 and P, do not yield practically useful estimators because of their inability to efficiently handle the log-partition function. Instead, we propose an estimator, hatpn based on minimizing the Fisher divergence, J(p0Vert p) between p0 and p∈ P, which involves solving a simple finite-dimensional linear system. When p0∈P, we show that the proposed estimator is consistent, and provide a convergence rate of n^-minleftfrac23,frac2β+12β+2right in Fisher divergence under the smoothness assumption that log p0∈R(Cβ) for some β≥ 0, where C is a certain Hilbert-Schmidt operator on H and R(Cβ) denotes the image of Cβ. We also investigate the misspecified case of p0notinP and show that J(p0Verthatpn)rightarrow infp∈PJ(p0Vert p) as nrightarrow∞, 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.