Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces

Abstract

We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space R N . Each measure in the product is a shifted and scaled copy of a reference probability measure on R 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.

Maximum a posteriori estimators in ℓp are well-defined for diagonal Gaussian priors

“…Agapiou et al (2018) studied the MAP estimator for Bayesian inversion with sparsity-promoting Besov priors. The connection between weak and strong modes was further explored in Lie and Sullivan (2018), and Ayanbayev et al (2021a, 2021b discussed stability and convergence of global weak modes using Γ-convergence. Recently, Lambley and Sullivan (2022) presented a perspective on modes via order theory.…”

“…When working in sequence spaces X ⊆ R N , such as ℓ p spaces, one important technique (Dashti et al 2013, Agapiou et al 2018, Ayanbayev et al 2021b is to consider…”

Γ -convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems

“…This section illustrates the preceding general theory of convergence of modes via Γconvergence of OM functionals by means of two key examples, namely Gaussian and Besov B s 1 measures, both of which commonly arise as prior distributions in BIPs. Besov B s p -priors with 1 p 2, Cauchy priors, and more general product measures are treated in a unified way in part II of this paper (Ayanbayev et al 2021). The convergence of modes (MAP estimators) for posterior distributions will be discussed in section 6.…”

“…Section 5 develops this idea in two prototypical settings, namely Gaussian and Besov-1 measures, which are frequently used as Bayesian prior distributions; these results can then be transferred to measures that are absolutely continuous with respect to these paradigmatic examples and can be interpreted as the corresponding posterior measures. More general prior measures, which include Cauchy measures and Besov-p measures for 1 p 2 are treated in part II of this paper (Ayanbayev et al 2021). In section 6, these ideas are then applied to the convergence and stability of MAP estimators for BIPs, for which Gaussian and Besov-1 measures are prototypical prior distributions.…”

Strong maximum a posteriori estimation in Banach spaces with Gaussian priors

“…While this article considers only Gaussian priors, MAP estimators have also been studied for Bayesian inverse problems with Besov and Cauchy priors (Agapiou et al 2018, Ayanbayev et al 2022b. Besov and Cauchy priors are typically constructed as product measures placing full mass on a Banach subspace of R ∞ , and the product structure of R ∞ makes finite-dimensional approximation arguments possible.…”