Optimizing Gaussian Process Kernels Using Nested Sampling and ABC Rejection for H(z) Reconstruction

TL;DR

This work uses Gaussian process regression to reconstruct the expansion history $H(z)$ from cosmic chronometer data, exploring six kernels in both redshift spaces $z$ and $\log(z+1)$, and comparing kernel performance with nested sampling (NS) and ABC rejection. It integrates CC data with BAO, SN Ia, and Planck priors, analyzing four GP configurations (Full versus Diagonal covariance in each space) and assessing Hubble constant inferences $H_0$. Key findings show that reconstructions in $\log(z+1)$ space are physically reasonable and often advantageous, while non-diagonal covariances can degrade performance with CC data; kernel preferences are highly task- and method-dependent, with NS and ABC sometimes giving opposite rankings. In the joint BAO+SN Ia+CMB analysis, NS yields $H_0$ values near the Planck result (reaffirming the Hubble tension), whereas ABC leans toward SH0ES-like values, emphasizing that kernel and inference framework choices introduce systematic uncertainties that should be marginalized in cosmological error budgets. The results advocate for task-specific kernel selection, potential kernel ensembles, and consensus approaches to robustly quantify $H_0$ and other expansion-history inferences in the era of precision cosmology.

Abstract

Recent cosmological observations have achieved high-precision measurements of the Universe's expansion history, prompting the use of nonparametric methods such as Gaussian processes (GP) regression. We apply GP regression for reconstructing the Hubble parameter using CC data, with improved covariance modeling and latest study in CC data. By comparing reconstructions in redshift space $z$ and transformed space $\log(z+1)$ , we evaluate six kernel functions using nested sampling (NS) and approximate Bayesian computation rejection (ABC rejection) methods and analyze the construction of Hubble constant $H_0$ in different models. Our analysis demonstrates that reconstructions in $\log(z+1)$ space remain physically reasonable, offering a viable alternative to conventional $z$ space approaches, while the introduction of nondiagonal covariance matrices leads to degraded reconstruction quality, suggesting that simplified diagonal forms may be preferable for reconstruction. These findings underscore the importance of task-specific kernel selection in GP-based cosmological inference. In particular, our findings suggest that careful preliminary screening of kernel functions, based on the physical quantities of interest, is essential for reliable inference in cosmological research using GP.

Figures (11)

• Figure 1: Evidence comparison across kernel functions for the four reconstruction models defined in Table \ref{['tab:model']}, obtained from nested sampling of the CC data. Error bars indicate the 1$\sigma$ uncertainties in the nested sampling estimates.
• Figure 2: In nested sampling: Bayes factor heatmaps comparing all pairs of kernel functions within each of the four reconstruction models, based on CC data. Each panel corresponds to one model, and the color scale indicates the strength of preference according to the Bayes factor. Heatmap values correspond to evidence ratios (horizontal/vertical).
• Figure 3: Reconstructed Hubble parameter $H(z)$ from cosmic CC data using nested sampling. Each panel corresponds to one of the four reconstruction models listed in Table \ref{['tab:model']}. Within each panel, six kernel functions are shown, reconstructed under their respective optimal hyperparameters in NS. The shaded bands around each reconstruction denote the 1$\sigma$ uncertainties of the Gaussian process predictions. The CC measurements with 1$\sigma$ error bars are plotted for reference.
• Figure 4: Comparison of the estimated Hubble constant $H_0$ across different kernel functions and reconstruction models using nested sampling. Different colors indicate the four reconstruction models. The error bars represent the 1$\sigma$ (68% confidence level) uncertainties from the posterior distributions.
• Figure 5: Normalized posterior probabilities of different kernel functions obtained from the ABC rejection analysis of the CC data, shown across the four reconstruction models in Table \ref{['tab:model']}. Colors indicate different models; the vertical axis shows their corresponding normalized probabilities. These values reflect the relative support for each kernel within a given model after normalization.