Does Cameron–Martin coercivity ensure a strong posterior mode?
Let $X$ be a real separable Banach space, $\mu_0$ a centred nondegenerate Gaussian Borel probability measure on $X$ with Cameron–Martin space $(E, \|\cdot\|_E)$, and $\Phi : X \to \mathbb{R}$ continuous with $$0 < Z := \int_X e^{-\Phi(x)}\, \mu_0(dx) < \infty, \qquad \mu(dx) := Z^{-1} e^{-\Phi(x)}\, \mu_0(dx).$$ For $h \in E$ let $I(h) := \Phi(h) + \tfrac12 \|h\|_E^2$ be the posterior Onsager–Machlup functional. With closed balls $B_r(x)$ in the norm of $X$ and $M_r := \sup_{x \in X} \mu(B_r(x))$, a strong mode of $\mu$ is a point $x_* \in X$ with $$\lim_{r \downarrow 0} \frac{\mu(B_r(x_*))}{M_r} = 1.$$ Lambley proved that $\mu$ has a strong mode if for every $\eta > 0$ there is $K(\eta) \in \mathbb{R}$ with $\Phi(x) \ge K(\eta) - \eta \|x\|_X^2$ for all $x \in X$, and that the coercivity condition below gives a weak mode. Klebanov, Lambley and Sullivan constructed continuous potentials on $\ell^2$ with a weak mode but no strong mode; their Onsager–Machlup functionals fail this coercivity condition.
Problem. Suppose there are constants $A \in \mathbb{R}$ and $c > 0$ such that $A + I(h) \ge c \|h\|_E^2$ for every $h \in E$. Must $\mu$ have a strong mode?
Source: Lambley (2023), §4.1, the question immediately before Proposition 4.1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- ST
- Keywords
- AIM Open Applied Problems, Gaussian measures, maximum a posteriori estimators, small-ball modes
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{lambley2023,
author = {Lambley, Hefin},
title = {Strong maximum a posteriori estimation in {B}anach spaces with {G}aussian priors},
journal = {Inverse Problems},
volume = {39},
number = {12},
year = {2023},
pages = {125010},
doi = {10.1088/1361-6420/ad07a4},
eprint = {2304.13622},
archivePrefix = {arXiv},
}
@misc{klebanov2025,
author = {Klebanov, Ilja and Lambley, Hefin and Sullivan, T. J.},
title = {Classification of small-ball modes and maximum a posteriori estimators in metric spaces},
year = {2025},
eprint = {2306.16278},
archivePrefix = {arXiv},
note = {Version 4, 14 August 2025},
}
@misc{aim-does-cameronmartin-coercivity-ensure-a-strong-posterior-mode,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does Cameron–Martin coercivity ensure a strong posterior mode?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/667-does-cameronmartin-coercivity-ensure-a-strong-posterior-mode.md},
note = {Entry 667 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01114, "Does Cameron–Martin coercivity ensure a strong posterior mode?", clopen.solutions.