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

Abstract

We prove that maximum a posteriori estimators are well-defined for diagonal Gaussian priors µ on lp under common assumptions on the potential Φ. Further, we show connections to the Onsager–Machlup functional and provide a corrected and strongly simplified proof in the Hilbert space case p = 2, previously established by Dashti et al. (2013); Kretschmann (2019).
These corrections do not generalize to the setting 1 ≤ p < ∞, which
requires a novel convexification result for the difference between the Cameron–Martin norm and the p-norm.

Strong maximum a posteriori estimation in Banach spaces with Gaussian priors

“…Then, the bound in proposition 3.1 is used to show that any asymptotic maximising family (definition 4.2) for the posterior has a limit point (lemma 4.4). Lemma 4.5 shows that such a point must be a strong mode, extending a previous proof of Klebanov and Wacker (2023) to the Banach case, and this completes the proof of theorem 1.1.…”

“…We now prove the main theorem on the existence of strong MAP estimators. The strategy of the proof is similar in spirit to the prior work of Dashti et al (2013), Kretschmann (2019Kretschmann ( , 2023 and Klebanov and Wacker (2023).…”

“…Finally, this limit point is shown to be both a strong MAP estimator and an OM minimiser. Klebanov and Wacker (2023) point out that it is not obvious that the radius-r maximisers exist and show that the proof can be adapted to use a family (x r ) r>0 of 'approximate maximisers' nearly attaining the supremal radius-r mass instead. Klebanov and Wacker (2023) call such a family an asymptotic maximising family.…”

“…As pointed out by Klebanov and Wacker (2023), although Dashti et al (2013) stated their results in the Banach setting, technical complications limit their proof strategy to Hilbert spaces, and some additional results are needed to complete the proof even in the Hilbert case, as described by Kretschmann (2019).…”

“…The conditions imposed on the potential are weaker than those used by Dashti et al (2013) and Klebanov and Wacker (2023), who assumed that the potential was globally bounded below and locally Lipschitz. As pointed out by Kretschmann (2019Kretschmann ( , 2023, a global lower bound on Φ excludes Bayesian inverse problems with observations corrupted by additive white noise or Laplacian noise, and the proof of Dashti et al (2013) can be extended to handle these cases using a less restrictive lower bound.…”