Abstract
We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space . Each measure in the product is a shifted and scaled copy of a reference probability measure on that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.
Highlights
A maximum a posteriori (MAP) estimator is an important feature of a Bayesian inverse problem (BIP) because of its interpretation as a mode of the posterior distribution, i.e. as a point in parameter space X to which the posterior assigns the most mass, relative to other points.This interpretation is only heuristic, because even in the straightforward case that the parameter space has finite dimension and the posterior admits a Lebesgue density, every point will have measure zero
We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class
Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory
Summary
A maximum a posteriori (MAP) estimator is an important feature of a Bayesian inverse problem (BIP) because of its interpretation as a mode of the posterior distribution, i.e. as a point in parameter space X to which the posterior assigns the most mass, relative to other points. The first main contribution of this paper, theorem 4.10, shows the existence of and derives an explicit formula for OM functionals of measures in this class under another technical assumption. The second main contribution is to prove equicoercivity and Γ-convergence of OM functionals associated to a convergent sequence in this class, where convergence is meant in the sense of convergence of the scale and shift sequences, and convergence of the Lebesgue densities of the reference probability measures: see theorems 4.13 and 4.14. We collect auxiliary results in appendix A and state technical proofs in appendix B
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