Log-density estimation in linear inverse problems

Abstract

We estimate a probability density function p which is related by a linear operator K to a density function q in sequences of regular exponential families based on a random sample from q. In this paper deconvolution and positron emission tomography are considered. The logarithm of the density function is approximated by basis functions consisting of singular functions of K. While direct maximum likelihood (or minimum Kullback-Leibler) density estimation in exponential families selects the parameters to match the moments of the basis functions to the sample moments, in the inverse problem the moment of each singular function is related to a corresponding moment of the direct problem by a factor given by a singular value $\lambda_{\nu}$ of K. Thus an appropriate analogue of the maximum likelihood estimate is obtained by matching moments with respect to p to $l/\lambda_{\nu}$ times the empirical moments associated with the sample from q. Bounds on the Kullback-Leibler distance between the true density and the estimators are obtained and rates of convergence are established for log-density functions having a measure of smoothness. The density estimator converges to the unknown density in the Kullback-Leibler sense and in the $L_2$-sense at a rate determined not only by the order of smoothness of the log-density and the dimension of data but also by the decay rate of the singular values of the operator. A minimax lower bound for deconvolution is provided under certain conditions. Numerical examples using simulated data are provided to illustrate the finite-sample performance of the proposed method for deconvolution and positron emission tomography.