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Bayesian Model Selection with an Application to Cosmology

Nikoloz Gigiberia

TL;DR

This work applies Bayesian inference to cosmology using DES-SN5YR Type Ia supernovae to compare ΛCDM, wCDM, and CPL models. It combines Hamiltonian Monte Carlo with the No-U-Turn Sampler and bridge sampling to estimate posterior distributions and Bayes factors, respectively, alongside WAIC for predictive evaluation. The results show nearly identical predictive performance across models (per WAIC), but Bayes factors decisively favour a constant-$w$ model ($w$CDM) over ΛCDM and CPL, with ΛCDM preferred over CPL. These findings suggest that, for DES-SN5YR data, allowing a freely varying constant equation of state improves model balance between fit and parsimony, while strong evolution in $w(z)$ (CPL) is not supported.

Abstract

We investigate cosmological parameter inference and model selection from a Bayesian perspective. Type Ia supernova data from the Dark Energy Survey (DES-SN5YR) are used to test the $Λ$CDM, $w$CDM, and CPL cosmological models. Posterior inference is performed via Hamiltonian Monte Carlo using the No-U-Turn Sampler (NUTS) implemented in NumPyro and analyzed with ArviZ in Python. Bayesian model comparison is conducted through Bayes factors computed using the bridgesampling library in R. The results indicate that all three models demonstrate similar predictive performance, but $w$CDM shows stronger evidence relative to $Λ$CDM and CPL. We conclude that, under the assumptions and data used in this study, $w$CDM provides a better description of cosmological expansion.

Bayesian Model Selection with an Application to Cosmology

TL;DR

This work applies Bayesian inference to cosmology using DES-SN5YR Type Ia supernovae to compare ΛCDM, wCDM, and CPL models. It combines Hamiltonian Monte Carlo with the No-U-Turn Sampler and bridge sampling to estimate posterior distributions and Bayes factors, respectively, alongside WAIC for predictive evaluation. The results show nearly identical predictive performance across models (per WAIC), but Bayes factors decisively favour a constant- model (CDM) over ΛCDM and CPL, with ΛCDM preferred over CPL. These findings suggest that, for DES-SN5YR data, allowing a freely varying constant equation of state improves model balance between fit and parsimony, while strong evolution in (CPL) is not supported.

Abstract

We investigate cosmological parameter inference and model selection from a Bayesian perspective. Type Ia supernova data from the Dark Energy Survey (DES-SN5YR) are used to test the CDM, CDM, and CPL cosmological models. Posterior inference is performed via Hamiltonian Monte Carlo using the No-U-Turn Sampler (NUTS) implemented in NumPyro and analyzed with ArviZ in Python. Bayesian model comparison is conducted through Bayes factors computed using the bridgesampling library in R. The results indicate that all three models demonstrate similar predictive performance, but CDM shows stronger evidence relative to CDM and CPL. We conclude that, under the assumptions and data used in this study, CDM provides a better description of cosmological expansion.

Paper Structure

This paper contains 31 sections, 7 theorems, 68 equations, 13 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Cosmological Theory
  3. Model Assumptions
  4. Cosmological Models
  5. LambdaCDM Model
  6. wCDM Model
  7. CPL Model
  8. The Distance Modulus
  9. Data
  10. Models and Methodology
  11. Bayesian Setup
  12. Hamiltonian Monte Carlo
  13. The NUTS Sampler
  14. Prior Choice
  15. Cholesky Whitening Transformation
  16. ...and 16 more sections

Key Result

Lemma 4.1

Let $L$ be the (lower-triangular) Cholesky factor of $\Sigma$, i.e. $\Sigma=LL^\top$, and define $T:=L^{-1}$. For the likelihood satisfies where $\Phi_n(\,\cdot\,;0,I)$ is the $n$-variate standard normal density and $\phi$ is the univariate standard normal density.

Figures (13)

  • Figure 1: The Hubble Diagram for the DES-SN5YR dataset.
  • Figure 2: The path taken by a Metropolis-Hastings and Hamiltonian Monte Carlo MCMC on a bivariate Gaussian distribution with 200 samples drawn.
  • Figure 3: Trace plots for posterior samples of $H_0$ and $\Omega_m$under the $\Lambda CDM$ model. The two set of curves on the left show the posteriors estimated by four independent HMC chains. Each chain shows stable mixing and no visible divergences in the trace plots on the right, indicating good convergence of the NUTS sampler in NumPyro.
  • Figure 4: Same as Fig. 3, but for the $wCDM$ model. The chains again show good convergence, with slight skewing for the $w$ parameter's trace.
  • Figure 5: Same as Fig. 3, but for the $CPL$ model. The chains again show good convergence, with skewness for the $\Omega_m$ parameter's trace and similar skewing for the $w_a$ parameter as in Fig. 4 for $w$.
  • ...and 8 more figures

Theorems & Definitions (15)

  • Lemma 4.1: Exact likelihood reparameterisation
  • proof
  • Lemma 4.2: Posterior invariance under whitening
  • proof
  • Theorem 4.3: MH invariance under whitening
  • proof
  • Theorem 4.4: HMC flow invariance under whitening
  • proof
  • Theorem 4.5: Equivariance of Leapfrog under Affine Transformations
  • proof
  • ...and 5 more