High‑probability conditional likelihood bound for large‑support Poisson priors

Establish that, in the Poisson empirical Bayes setting with true prior G0 supported on [0, A] where A = A_n may grow up to O(n^2), and with a training prior‑on‑prior that draws k = \widetilde{\Theta}(\sqrt{A}) atom locations uniformly on [0, A] with Dirichlet(1,…,1) weights, the training conditional distribution Π_{X_n|X^{n−1}} satisfies −log Π_{X_n|X^{n−1}} = O(polylog(n)) with probability at least 1 − (n\sqrt{A})^{−1} over X^n ∼ f_{G0}^{\otimes n}. Proving this bound would imply, together with known metric entropy and Hellinger–regret inequalities, that the pretrained estimator achieves the minimax regret rate \widetilde{\Theta}(A^{1.5}/n) for A = O(n^2).

Background

The paper extends its universal‑prior framework beyond compact support and shows results for subexponential priors. It further discusses the regime where the prior support size A = A_n grows and references known optimal regret rates of order \widetilde{\Theta}(A^{1.5}/n) for A up to O(n^2).

The authors argue that their pretrained approach with k ≍ \sqrt{A} atoms should recover this regret rate if one can prove a specific high‑probability lower bound on the training conditional likelihood Π{X_n|X^{n−1}}. This bound, phrased as −log Π{X_n|X^{n−1}} = O(polylog(n)) with high probability, aligns with parameters in existing Hellinger‑to‑regret inequalities and metric entropy estimates. They were unable to prove this bound and explicitly pose it as a conjecture.