This paper introduces a way to cheaply rank candidate solutions in Bayesian inverse problems by looking at how a small amount of likelihood information changes the prior’s moment geometry. The authors build a likelihood-weighted moment matrix and then use two classical constructs — the Christoffel ratio and a Nevai score — to turn that matrix into inexpensive relevance scores. The goal is not to replace the exact likelihood, but to reuse a modest “pilot” of likelihood evaluations to screen or reorder many more candidates without solving the forward model each time.

At a high level the method records how multiplying the prior by the likelihood alters polynomial moments of the prior measure. Those moments form a finite matrix that the authors call a posterior-weighted Hankel moment matrix. The Christoffel ratio compares local moment mass before and after weighting and highlights where the posterior puts relatively more mass. The Nevai score is a related, linear, and numerically stable polynomial approximation of the likelihood. Both scores can be evaluated at a new candidate only by computing polynomial features and a low-dimensional matrix quadratic form; the Christoffel score also benefits from a one-time matrix factorization.

The moment matrix can be built in several ways. It can come from deterministic likelihood evaluations at a set of prior draws (the pilot stage). It can also be estimated from bounded unbiased likelihood marks, from posterior samples, or even from unbiased binary likelihood observations. For function-valued unknowns the authors propose nested feature maps. At each feature level the conditional likelihood is the exact Bayesian update visible through those features; this preserves evidence and converges to the full posterior as the feature resolution grows.

Why this matters: many inverse problems are costly because each candidate requires an expensive forward solve. Here a pilot of size M can be used to rank a much larger pool of N candidates (the intended regime has M much smaller than N). That can focus expensive forward work on a small subset of promising candidates. The learned moment geometry can also be used in a different mode that does not approximate the posterior: by reordering exact stochastic likelihood blocks one can reduce expected computational work without changing the final posterior.