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

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 $1\le p<\infty$, which requires a novel convexification result for the difference between the Cameron--Martin norm and the $p$-norm.

References in corpus (3)

• Γ-convergence of Onsager-Machlup functionals. Part I: With applications to maximum a posteriori estimation in Bayesian inverse problems
• Γ-convergence of Onsager-Machlup functionals. Part II: Infinite product measures on Banach spaces
• An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems

Cited by in corpus (3)

• An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems
• Are minimizers of the Onsager-Machlup functional strong posterior modes?
• Strong maximum a posteriori estimation in Banach spaces with Gaussian priors