On the Onsager--Machlup functional of the infinite-dimensional angular central Gaussian distribution and related measures

Abstract

The Onsager--Machlup functional of a probability measure is a generalisation of the negative logarithm of the probability density function, defined precisely using the ratios of masses of balls in the limit of vanishing radius. It plays an important role in several areas of applied mathematics because its minimisers can be understood as modes of the measure. We show that the Onsager--Machlup functional of the angular central Gaussian measure on the unit sphere of an infinite-dimensional Hilbert space---i.e., the radial projection of a centred Gaussian measure with covariance operator $C$---is constant on the finite-dimensional sphere spanned by the eigenvectors of leading eigenvalue under $C$, and takes the value $+\infty$ elsewhere. We discuss the consequences of this extreme concentration of mass for maximum a posteriori estimation in Bayesian inverse problems with an angular central Gaussian prior.