Partition function approach to non-Gaussian likelihoods: formalism and expansions for weakly non-Gaussian cosmological inference

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 determination confidence contours. As an alternative, we construct canonical partition functions as Laplace-transforms Bayesian evidence, from which MCMC-methods would sample microstates. Cumulants order $n$ posterior distribution follow by $n$-fold differentiation logarithmic function, recovering classic Fisher-matrix formalism at second order. We connect this approach weakly non-Gaussianities to DALI- and Gram-Charlier expansions demonstrate validity with supernova-likelihood on cosmological parameters $\Omega_m$ $w$. comment extensions function include kinetic energies bridge Hamilton sampling, ensemble methods, they result transitioning macrocanonical depending chemical potential. Lastly relationship Cram\'er-Rao boundary information entropies.