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

TL;DR

This work develops a Markovian framework to explore the discrete space of competing cosmological descriptions, guided by Bayesian evidence, by introducing a two-layer setup with discrete models and continuous parameters and a Metropolis-Hastings-like sampler that yields posterior model probabilities without exhaustive evidence calculation. It connects model-space sampling to partition-function concepts, defining canonical partitions and thermodynamic-like quantities to quantify information gain and uncertainty in model selection, including a binary-key representation to encode polynomial hypotheses. The method is validated on a linear toy model and then applied to Pantheon+ Type Ia supernova data, where polynomial descriptions of the dark energy equation of state $w(a)$ are tested; the results robustly favor a constant $w$ near $-0.94$ (MAP $p\approx0.93$) with weaker evidence for higher-order terms, and the conclusions show limited sensitivity to priors in well-constrained regions. Overall, the approach offers an efficient, principled way to perform Bayesian model selection across large model spaces and to quantify model-uncertainty through posterior probabilities and related information measures.

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 space. In analogy to the logarithmic likelihood ratio in parameter estimation problem, the process is governed by the logarithmic evidence ratio. We apply our methodology to selecting a polynomial for the dark energy equation of state function $w(a)$ on the basis of data for the supernova distance-redshift relation.

Figures (7)

• Figure 1: Conceptual overview of the two different levels of inference. For each model $M_i$ in the discrete space, a corresponding continuous parameter space exists. The discrete pdf for all models is obtained with the evidence for each model and its prior.
• Figure 2: Outline of the discrete two-dimensional model space of polynomials.
• Figure 3: Two data sets are produced from the same model with small (brown) and large (teal) noise. The true model is of the form $y=\theta_0 + \theta_1 x + \theta_3 x^3$. The lines in the respective colours show the best-fit model obtained from the model posterior. For small noise, the best fit and truth coincide with a probability of 0.84. For large noise, a constant model is the most likely, yet with a probability of 0.75, as the effect from higher order terms is washed away by the noise.
• Figure 4: Toy model evaluation: (Left) Comparison of different noise levels in generated data - the less noise, the better the model selection. (Centre) Variety of model priors compared - stronger penalisation of more complex models performs better. (Right) For a Gaussian parameter prior different covariances were tested - a more narrow prior for suitable expectation value results into lower NRMSE (Normalised Root Mean Square Error). The drawn lines are the quantiles of the corresponding data set with the line in the centre being the median.
• Figure 5: The three figures show the effect of four sensible prior choices for the model parameters on the model posterior. Values smaller than $10^{-3}$ are cut off. (Left) Each bar is the probability that the term of the polynomial is present in the model, independent of the others. (Centre) The marginalised posterior such that the histogram shows the probability distribution for the polynomial degree and (right) similarly for the degree of freedom.