Partition function approach to non-Gaussian likelihoods: partitions for the inference of functions and the Fisher-functional

Abstract

ABSTRACT Motivated by constraints on the dark energy equation of state from a data set of supernova distance moduli, we propose a formalism for the Bayesian inference of functions: Starting at a functional variant of the Kullback–Leibler divergence we construct a functional Fisher-matrix and a suitable partition functional which takes on the shape of a path integral. After showing the validity of the Cramér–Rao bound and unbiasedness for functional inference in the Gaussian case, we construct Fisher-functionals for the dark energy equation of state constrained by the cosmological redshift–luminosity relationship of supernovae of type Ia, for both the linearized and the lowest-order nonlinear models. Introducing Fourier-expansions and expansions into Gegenbauer polynomials as discretizations of the dark energy equation of state function shows how the uncertainty on the inferred function scales with model complexity and how functional assumptions can lead to errors in extrapolation to poorly constrained redshift ranges.