Hankel--Christoffel--Nevai Screening for Bayesian Inverse Problems

Abstract

Likelihood evaluation in Bayesian inverse problems often requires a forward-model solve. We study candidate screening using a likelihood-weighted prior moment matrix and two associated scores: a Christoffel ratio and a Nevai polynomial average. A finite pilot supplies likelihood information that is reused to rank further prior candidates. We connect finite-pilot matrix error, quantitative Legendre localization, and fixed-budget posterior-mass regret. Conditional likelihoods distinguish feature loss from polynomial and sampling errors, and a filtered Gaussian construction permits controlled feature selection and localization. In a nonlinear function-coefficient PDE, equal-budget comparisons separate methods using scalar likelihoods from surrogates using forward outputs. The Christoffel ratio outperforms the tested direct likelihood regressions, but forward-response surrogates capture more posterior mass at small retention budgets. Independent-pilot tests and reference resampling quantify two distinct sources of uncertainty. Controlled Gaussian experiments illustrate finite-filter truncation and the benefit of selecting relevant features. These results identify when moment geometry provides useful screening information, without asserting universal superiority over surrogate models or an exact posterior sampling method.