Posterior Alternative Calibration in Ambiguous Inverse Problems

Ambiguous inverse problems often admit multiple clean signals that are consistent with the same observation. In such settings, a sampler should not only produce plausible or measurement-consistent reconstructions, but also allocate probability mass correctly across the alternatives that remain unresolved. We formalize this requirement as posterior alternative calibration and introduce mode coverage error (MCE), a finite-alternative diagnostic that measures calibration after pushing the posterior and sampler distributions onto a benchmark-specified alternative label space. This perspective reveals a basic failure mode of point-centered inverse solvers: measurement consistency can be preserved by averaging across alternatives, even when the resulting point lies outside the nonconvex set of valid posterior explanations. We prove that local point-denoising updates can therefore suffer barycentric collapse under weak branch identification, while calibrated local branch probabilities are sufficient to control MCE. Across controlled Gaussian, imbalanced, nullspace, and template-inpainting diagnostics, MCE exposes posterior mass-allocation failures that are missed by residual error, recall, diversity, and confidence. A semantic-label inpainting benchmark with DDIM, DPS, DDNM, RePaint, and clustering-based samplers further shows that practical samplers can recover all specified alternatives while assigning different branch probabilities. These results position MCE as a targeted evaluation axis for finite-alternative ambiguity benchmarks, complementary to reconstruction fidelity and sample quality.