We introduce a Hankel--Christoffel--Nevai framework for screening posterior-relevant candidates in Bayesian inverse problems. A likelihood-weighted moment matrix records how Bayesian updating changes the geometry of the prior, and Christoffel and Nevai constructions convert this information into inexpensive relevance scores. The Christoffel ratio captures relative local moment mass, whereas the Nevai score provides a stable polynomially localized approximation of the likelihood. The moment matrix can be estimated from deterministic likelihood values, bounded unbiased marks, posterior samples, or binary unbiased likelihood observations. The construction extends to function-valued unknowns through nested feature maps, where the induced conditional likelihood has an exact Bayesian interpretation. We establish consistency and error bounds that separate feature resolution, polynomial localization, and pilot estimation. The same learned geometry can also be used to reorder exact stochastic likelihood blocks, reducing expected work without altering the posterior target. Numerical experiments on nonlinear, PDE-based, and function-space inverse problems demonstrate effective posterior-mass screening and computational savings.
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