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) and 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.