Quasi-Bayes empirical Bayes: a sequential approach to the Poisson compound decision problem

The Poisson compound decision problem is a long-standing problem in statistics, where empirical Bayes methodologies are commonly used to estimate Poisson's means in static or batch domains. In this paper, we study the Poisson compound decision problem in a streaming or online domain. Adopting a quasi-Bayesian approach, referred to as Newton's algorithm, we obtain a sequential estimate that is easy to evaluate, computationally efficient, and maintain a constant per-observation computational cost as data accumulate. Asymptotic frequentist guarantees of this estimate are established, showing consistency and asymptotic optimality, where the latter is understood as vanishing excess Bayes risk or regret. We demonstrate the effectiveness of our methodology through empirical analysis on synthetic and real data, with comparisons to existing approaches.