Markov Walk Exploration of Model Spaces: Bayesian Selection of Dark Energy Models with Supernovae

Markov Walk Exploration of Model Spaces: Bayesian Selection of Dark Energy Models with Supernovae
Benedikt Schosser, Tobias R"ospel, Bjoern Malte Schaefer
arXiv:2407.06259v1 Announce Type: new
Abstract: Central to model selection is a trade-off between performing a good fit and low model complexity: A model of higher complexity should only be favoured over a simpler model if it provides significantly better fits. In Bayesian terms, this can be achieved by considering the evidence ratio, enabling choices between two competing models. We generalise this concept by constructing Markovian random walks for exploring the entire model spaces governed by the logarithmic evidence ratio, in analogy to the logarithmic likelihood ratio in parameter estimation problems. The theory of Markovian model exploration has an analytical description with partition functions, which we derive for both the canonical and macrocanonical case. We apply our methodology to selecting a polynomial for the dark energy equation of state function $w(a)$ fulfilling sensible physical priors, on the basis of data for the supernova distance-redshift relation. We conclude by commenting on the Jeffrey scale for Bayesian evidence ratios, choices of model priors and derived quantities like Shannon entropies for posterior model probabilities.arXiv:2407.06259v1 Announce Type: new
Abstract: Central to model selection is a trade-off between performing a good fit and low model complexity: A model of higher complexity should only be favoured over a simpler model if it provides significantly better fits. In Bayesian terms, this can be achieved by considering the evidence ratio, enabling choices between two competing models. We generalise this concept by constructing Markovian random walks for exploring the entire model spaces governed by the logarithmic evidence ratio, in analogy to the logarithmic likelihood ratio in parameter estimation problems. The theory of Markovian model exploration has an analytical description with partition functions, which we derive for both the canonical and macrocanonical case. We apply our methodology to selecting a polynomial for the dark energy equation of state function $w(a)$ fulfilling sensible physical priors, on the basis of data for the supernova distance-redshift relation. We conclude by commenting on the Jeffrey scale for Bayesian evidence ratios, choices of model priors and derived quantities like Shannon entropies for posterior model probabilities.