An essential convex hull criterion for false confidence in location models

Unrefereed research candidate. Balch, Martin and Ferson (2019) showed that for any additive probability produced from data, some false hypotheses tend to receive high probability, a phenomenon they called false confidence. Martin (2024) asked which hypotheses are affected. This candidate answers that question exactly for flat-prior posteriors in location models with radially decreasing errors.

Hull criterion. Let C_H be the closed convex hull of the essential support of a hypothesis H. For exponential-power errors with p>1, including the Gaussian case, H is valid at every threshold at a parameter exactly when that parameter is not in the interior of C_H. For a radial density g with tail probability S, this equivalence holds for all hypotheses if and only if liminf S(R)/g(R-b)=0 for every b>0. Tails like exp(-r log r) satisfy this in dimension one and fail it in every higher dimension.

Complements and marginals. A hypothesis and its complement are both globally valid only if the hypothesis is almost everywhere a halfspace. This corrects, in these models, a conjecture of Martin (2024): a ball is non-linear yet valid at every false parameter. The posterior of a real function is exactly calibrated at every parameter if and only if its sublevel sets are almost everywhere parallel halfspaces, that is, the function is a strictly monotone function of one linear form.

High confidence at a fixed level. All-threshold failure may occur only at low thresholds, so the hull criterion alone says nothing about 95% confidence: (-2,-1) union (1,2) in the standard Gaussian model never receives posterior mass above 0.39. With log-concave errors, if a closed convex set contains the complement of H and has posterior mass at most s<1/2 when the observation equals the truth, then H receives posterior probability at least 1-s with sampling probability at least one half. In a worked satellite conjunction example with sigma/R=10, a certain collision produces a computed collision probability of at most 0.01 with probability one.

Prior work and limits: the sufficient direction is elementary, halfspace validity is in Martin (2023), and the Gaussian converse combines classical Laplace-transform growth with a tail comparison. The model assumes known shape and exact matching of sampling and posterior error laws. The prior-art finding is a record of dated searches, not proof of priority. The archive contains the manuscript (PDF and Markdown source), three deterministic check scripts with retained output, the figure code and SHA-256 sums. Assurance: written proofs; internal AI-assisted review only (not human peer review, independent reproduction or proof-assistant verification). Creator: Anonymous. Text CC0-1.0; code MIT.