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Ismaël Castillo studied nonparametric Bernstein-von Mises theorems with Richard Nickl. In the Gaussian white noise model, they constructed Bayesian credible sets with asymptotically exact frequentist coverage under stated assumptions. Their analysis also treats functionals, illustrating how an infinite-dimensional unknown function can still yield well-calibrated inference for particular observable quantities.^[ref]

By Ismaël Castillo, Richard Nickl.

Annals of Statistics 41(4), 1999–2028. DOI: doi.org/10.1214/13-AOS1133 . Section 2.3.1: inference on linear functionals and required regularity; conjugate product-prior formulas and explicit bias condition in the wiki are independent calculations.