A Sharp Prior-Sensitivity Threshold for Fixed-Partition Support Boundaries

Abstract

Consider a Poisson support-boundary experiment with intensity scale n, a known partition of [0,1] into K bins of widths Delta_j, and independent prior densities on the bin heights. For a fixed positive prior whose logarithm is locally Lipschitz, we prove that the posterior is asymptotically equivalent in total variation to the known product-exponential reference whenever sum_j (n Delta_j)^(-2) tends to zero, uniformly over bounded true heights. On an equal grid this gives K=o(n^(2/3)). A finite-sample bound follows by tensorizing squared coordinate errors. A fixed proper Laplace prior proves sharpness for this prior class: if L^2 K^3/n^2 tends to tau, the limiting total variation distance is 2 Phi(sqrt(tau)/2)-1, with distance tending to one above the critical scale under the stated localization condition. We also give the resulting integral-posterior approximation and an exact one-bin correction characterization. The exponential reference and the asymptotic K/n bias correction are due to earlier work; our result concerns their range of validity for fixed, known bins. This is an unrefereed, self-audited preprint with AI assistance, not a claim of whole-source closure or certified absolute priority.