  Motivated by constraints on the dark energy equation of state from
supernova-data, 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
Cram\'er-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 linearised and the lowest-order non-linear model.
Introducing Fourier-expansions and expansions into Gegenbauer-polynomials as
discretisations 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.
