Optimal inferential models for a Poisson mean

Statistical inference on the mean of a Poisson distribution is a fundamentally important problem with modern applications in, e.g., particle physics. The discreteness of the Poisson distribution makes this problem surprisingly challenging, even in the large-sample case. Here we propose a new approach, based on the recently developed framework of inferential models (IMs). Specifically, we construct optimal, or at least approximately optimal, IMs for two important classes of assertions/hypotheses about the Poisson mean. For point assertions, we develop a novel recursive sorting algorithm to construct this optimal IM. Numerical comparisons of the proposed method to existing methods are given, for both the mean and the more challenging mean-plus-background problem.

💡 Research Summary

Statistical inference for the mean μ of a Poisson distribution is a classic problem that resurfaces in modern high‑energy physics, epidemiology, and quality control. The discrete nature of the Poisson law makes the construction of confidence intervals and hypothesis tests surprisingly delicate, especially when the true mean is small or when a known background component is present. Traditional approaches—exact Clopper‑Pearson intervals, normal‑approximation score intervals, or Bayesian posterior credible intervals—either become overly conservative, lose nominal coverage, or depend on arbitrary prior choices.

The authors address these difficulties within the recently proposed inferential model (IM) framework. An IM starts by writing the data‑generating equation X = a(μ,U) with U∼Uniform(0,1). Uncertainty about the unobservable auxiliary variable U is then represented by a predictive random set (PRS). The PRS induces a plausibility function for any assertion about μ; validity of the IM is guaranteed when the PRS satisfies a simple stochastic ordering condition, ensuring that the resulting plausibility values are stochastically larger than uniform under the true parameter.

Two families of assertions are considered. For point assertions H₀: μ = μ₀, the paper introduces a novel recursive sorting algorithm that builds an optimal PRS. The algorithm orders possible observed counts x by their likelihood under μ₀, then incrementally expands the PRS in that order while preserving validity. At each step the algorithm selects the smallest PRS that still yields a valid IM, which is equivalent to minimizing the expected length of the resulting confidence interval. The authors prove that, under the criterion of minimal average length, the PRS produced by the recursive sorting algorithm is optimal among all valid IMs for point hypotheses.

For interval assertions μ∈