Indirect density estimation using the iterative Bayes algorithm

Many practical problems involve density estimation from indirect observations and they are classified as indirect density estimation problems. For example, image deblurring and image reconstruction in emission tomography belong to this class. In this paper we propose an iterative approach to solve these problems. This approach has been successfully applied to emission tomography (Ma, 2008). The popular EM algorithm can also be used for indirect density estimation, but it requires that observations follow Poisson distributions. Our method does not involve such assumptions; rather, it is established simply from the Bayes conditional probability model and is termed the Iterative Bayes (IB) algorithm. Under certain regularity conditions, this algorithm converges to the positively constrained solution minimizing the Kullback-Leibler distance, an asymmetric measure involving both logarithmic and linear scales of dissimilarities between two probability distributions.