Estimating multiple concurrent processes

We consider two related problems of estimating properties of a collection of point processes: estimating the multiset of parameters of continuous-time Poisson processes based on their activities over a period of time t, and estimating the multiset of activity probabilities of discrete-time Bernoulli processes based on their activities over n time instants. For both problems, it is sufficient to consider the observations´ profile - the multiset of activity counts, regardless of their process identities. We consider the profile maximum likelihood (PML) estimator that finds the parameter multiset maximizing the profile´s likelihood, and establish some of its competitive performance guarantees. For Poisson processes, if any estimator approximates the parameter multiset to within distance ε with error probability δ, then PML approximates the multiset to within distance 2ε with error probability at most δ · e^4√t·S, where S is the sum of the Poisson parameters, and the same result holds for Bernoulli processes. In particular, for the L₁ distance metric, we relate the problems to the long-studied distribution-estimation problem and apply recent results to show that the PML estimator has error probability e^-(t·S)0.9 for Poisson processes whenever the number of processes is k = O(tS log(tS)), and show a similar result for Bernoulli processes. We also show experimental results where the EM algorithm is used to compute the PML.