Maximum a posteriori estimators in $\ell^p$ are well-defined for diagonal Gaussian priors

TL;DR

This paper establishes the well-definedness of MAP estimators for diagonal Gaussian priors in ^p spaces, clarifies their connection to the Onsager--Machlup functional, and simplifies proofs in the Hilbert space case while addressing challenges for general p.

Contribution

It proves MAP estimators are well-defined for diagonal Gaussian priors in ^p and provides a simplified proof for the Hilbert space case, introducing a new convexification approach for general p.

Findings

MAP estimators are well-defined for diagonal Gaussian priors in ^p.

A simplified proof is provided for the Hilbert space case p=2.

A new convexification result addresses the p  case.

Abstract

We prove that maximum a posteriori estimators are well-defined for diagonal Gaussian priors $\mu$ on $\ell^p$ under common assumptions on the potential $\Phi$. 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) and Kretschmann (2019). These corrections do not generalize to the setting $1 \leq p < \infty$, which requires a novel convexification result for the difference between the Cameron--Martin norm and the $p$-norm.