Dashti–Stuart conjecture on MAP estimators for Banach-space Gaussian inverse problems

Let $X$ be the Banach space, let $\bar\beta$ be the Gaussian prior measure on $X$, let $\beta^y$ be the posterior measure associated with data $y$, let $E$ be the Cameron–Martin space, and let $\bar z$ be an element of $X$. An asymptotic maximizing family (AMF) for $\beta^y$ is a family of centers $\bigl(\bar z^4\bigr)_{4>0}$ maximizing the posterior mass of $4$-balls asymptotically as $4\to0$; an $X$-strong limit point is a limit in the norm of $X$ along a sequence $4_n\searrow0$. A MAP estimator is a limit point of such small-ball maximizing centers, and the OM functional is the Onsager–Machlup functional associated with the posterior.

Dashti–Stuart conjecture. Assume that the conditions in Assumption $4$ hold. Then:

1. The following statements are equivalent:

• $\bar z$ is an $X$-strong limit point as $4\to0$ of some AMF for $\beta^y$.
• $\bar z\in E$ and $\bar z$ minimizes the OM functional.
• $\bar z$ is a MAP estimator.

2. There exists at least one MAP estimator.

The conjecture extends the corresponding Hilbert-space result to Banach spaces. Its proof in the cited work is described as only partially correct in Hilbert spaces because the existence of $4$-ball maximizing centers is unclear; the Banach-space version remains open.