  Non-Gaussian likelihoods, ubiquitous throughout cosmology, are a direct
consequence of nonlinearities in the physical model. Their treatment requires
Monte-Carlo Markov-chain or more advanced sampling methods for the
determination of confidence contours. As an alternative, we construct canonical
partition functions as Laplace-transforms of the Bayesian evidence, from which
MCMC-methods would sample microstates. Cumulants of order $n$ of the posterior
distribution follow by direct $n$-fold differentiation of the logarithmic
partition function, recovering the classic Fisher-matrix formalism at second
order. We connect this approach for weakly non-Gaussianities to the DALI- and
Gram-Charlier expansions and demonstrate the validity with a
supernova-likelihood on the cosmological parameters $\Omega_m$ and $w$. We
comment on extensions of the canonical partition function to include kinetic
energies in order to bridge to Hamilton Monte-Carlo sampling, and on ensemble
Markov-chain methods, as they would result from transitioning to macrocanonical
partition functions depending on a chemical potential. Lastly we demonstrate
the relationship of the partition function approach to the Cram\'er-Rao
boundary and to information entropies.
