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

Abstract

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 Cramér-Rao boundary and to information entropies.

12 pages 2 figures, Affiliation updated

References in corpus (12)

• Improved Cosmological Constraints from New, Old and Combined Supernova Datasets
• Spectra and Light Curves of Six Type Ia Supernovae at 0.511 < z < 1.12 and the Union2 Compilation
• Bayes in the sky: Bayesian inference and model selection in cosmology
• Systematic Bias in Cosmic Shear: Beyond the Fisher Matrix
• Bayesian reconstruction of the cosmological large-scale structure: methodology, inverse algorithms and numerical optimization
• Present and future evidence for evolving dark energy
• Magnification-Temperature Correlation: the Dark Side of ISW Measurements
• Cosmological Systematics Beyond Nuisance Parameters : Form Filling Functions
• Biases on the cosmological parameters and thermal SZ residuals
• On Model Selection in Cosmology
• Fast calculation of the Fisher matrix for cosmic microwave background experiments
• Information geometry in cosmological inference problems

Cited by in corpus (1)

• Partition function approach to non-Gaussian likelihoods: macrocanonical partitions and replicating Markov-chains