Convex Optimization · Problem 7.7

Geometric problems — Boyd & Vandenberghe

Mathematics — Optimization · ml-estimation-poisson-distributions

Problem

ML estimation of Poisson distributions. Suppose xix_i, i=1,…,ni = 1, \ldots, n, are independent random variables with Poisson distributions prob(xi=k)=e−μiμikk!,\mathbf{prob}(x_i = k) = \frac{e^{-\mu_i} \mu_i^k}{k!}, with unknown means μi\mu_i. The variables xix_i represent the number of times that one of nn possible independent events occurs during a certain period. In emission tomography, for example, they might represent the number of photons emitted by nn sources. We consider an experiment designed to determine the means μi\mu_i. The experiment involves mm detectors. If event ii occurs, it is detected by detector jj with probability pjip_{ji}. We assume the probabilities pjip_{ji} are given (with pji≥0p_{ji} \geq 0, ∑j=1mpji≤1\sum_{j=1}^m p_{ji} \leq 1). The total number of events recorded by detector jj is denoted yjy_j, yj=∑i=1nyji,j=1,…,m.y_j = \sum_{i=1}^n y_{ji}, \quad j = 1, \ldots, m. Formulate the ML estimation problem of estimating the means μi\mu_i, based on observed values of yjy_j, j=1,…,mj = 1, \ldots, m, as a convex optimization problem. Hint. The variables yjiy_{ji} have Poisson distributions with means pjiμip_{ji}\mu_i, i.e., prob(yji=k)=e−pjiμi(pjiμi)kk!.\mathbf{prob}(y_{ji} = k) = \frac{e^{-p_{ji}\mu_i}(p_{ji}\mu_i)^k}{k!}. The sum of nn independent Poisson variables with means λ1,…,λn\lambda_1, \ldots, \lambda_n has a Poisson distribution with mean λ1+⋯+λn\lambda_1 + \cdots + \lambda_n.

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