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Revisiting Gaussian Process Reconstruction for Cosmological Inference: The Generalised GP (Gen GP) Framework

Ruchika, Purba Mukherjee, Arianna Favale

TL;DR

This work tackles kernel-induced biases in Gaussian Process reconstructions of the cosmic expansion history and $H_0$ estimation by introducing the Generalized Gaussian Process (Gen GP), which treats the Matérn smoothness ν as a free parameter to be inferred from data. By applying Gen GP to cosmic chronometer measurements and comparing against standard GP and parametric cosmologies (ΛCDM, wCDM, CPL, Padé), the authors show that kernel choice can produce variations in $H_0$ comparable to those from different cosmological models, and that full Bayesian marginalization over hyperparameters yields more robust, consistent reconstructions, especially when cosmological priors are included via the mean function. Gen GP achieves tighter and more reliable uncertainty estimates, reduces discrepancies between marginalized and optimized reconstructions, and mitigates kernel-selection bias without sacrificing fit quality. These results advocate for data-driven kernel optimization and comprehensive marginalization as essential for robust, model-independent cosmological inferences, a necessity as precision cosmology enters the data-rich era of Euclid and LSST.

Abstract

We investigate uncertainties in the estimation of the Hubble constant ($H_0$) arising from Gaussian Process (GP) reconstruction, demonstrating that the choice of kernel introduces systematic variations comparable to those arising from different cosmological models. To address this limitation, we introduce the Generalized Gaussian Process (Gen GP) framework, in which the Matérn smoothness parameter $ν$ is treated as a free parameter, allowing for data-driven kernel optimization. Using the cosmic chronometer Hubble data, we find that while standard GP with $Λ$CDM mean function exhibits noticeable reconstruction differences between optimized and marginalized approaches, particularly at $z > 1$, Gen GP maintains methodological consistency. In Gen GP, slight increases in $χ^2$ per degree of freedom relative to standard GP, for both the zero-mean and $Λ$CDM prior mean cases, reflect added flexibility rather than performance degradation. Our results emphasize that robust cosmological inference requires treating kernel parameters as free variables and implementing full Bayesian marginalization to avoid artificial precision from fixed hyperparameters. As machine learning becomes central to cosmological discovery, the Gen GP framework provides a principled approach to model-independent inference that properly accounts for methodological uncertainties while maintaining necessary flexibility for reliable parameter estimation.

Revisiting Gaussian Process Reconstruction for Cosmological Inference: The Generalised GP (Gen GP) Framework

TL;DR

This work tackles kernel-induced biases in Gaussian Process reconstructions of the cosmic expansion history and estimation by introducing the Generalized Gaussian Process (Gen GP), which treats the Matérn smoothness ν as a free parameter to be inferred from data. By applying Gen GP to cosmic chronometer measurements and comparing against standard GP and parametric cosmologies (ΛCDM, wCDM, CPL, Padé), the authors show that kernel choice can produce variations in comparable to those from different cosmological models, and that full Bayesian marginalization over hyperparameters yields more robust, consistent reconstructions, especially when cosmological priors are included via the mean function. Gen GP achieves tighter and more reliable uncertainty estimates, reduces discrepancies between marginalized and optimized reconstructions, and mitigates kernel-selection bias without sacrificing fit quality. These results advocate for data-driven kernel optimization and comprehensive marginalization as essential for robust, model-independent cosmological inferences, a necessity as precision cosmology enters the data-rich era of Euclid and LSST.

Abstract

We investigate uncertainties in the estimation of the Hubble constant () arising from Gaussian Process (GP) reconstruction, demonstrating that the choice of kernel introduces systematic variations comparable to those arising from different cosmological models. To address this limitation, we introduce the Generalized Gaussian Process (Gen GP) framework, in which the Matérn smoothness parameter is treated as a free parameter, allowing for data-driven kernel optimization. Using the cosmic chronometer Hubble data, we find that while standard GP with CDM mean function exhibits noticeable reconstruction differences between optimized and marginalized approaches, particularly at , Gen GP maintains methodological consistency. In Gen GP, slight increases in per degree of freedom relative to standard GP, for both the zero-mean and CDM prior mean cases, reflect added flexibility rather than performance degradation. Our results emphasize that robust cosmological inference requires treating kernel parameters as free variables and implementing full Bayesian marginalization to avoid artificial precision from fixed hyperparameters. As machine learning becomes central to cosmological discovery, the Gen GP framework provides a principled approach to model-independent inference that properly accounts for methodological uncertainties while maintaining necessary flexibility for reliable parameter estimation.

Paper Structure

This paper contains 20 sections, 18 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The Dual Nature Inference: Cosmological Models and Statistical Methods
  3. Cosmological Framework
  4. Statistical Framework: Gaussian Processes
  5. Kernel Selection
  6. Prior Mean Function Selection
  7. GP Methodology: Hyperparameter Sampling and Likelihood Evaluation
  8. Dataset Choice and Methodological Focus
  9. Parameter Inference and Sampling Strategy
  10. Bayesian Framework & Likelihood Evaluation
  11. Cosmological Model Analysis
  12. Implementation Details
  13. Gaussian Process Prediction Framework
  14. Analysis and Interpretation of Results
  15. Kernel vs Cosmology: A Comparative Study
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Reconstructed evolution of the Hubble parameter $H(z)$ using GP with a Matérn $\nu = 3/2$ kernel (top panel). The black dotted curve denotes the Planck best-fit $\Lambda$CDM model. The blue and red shaded regions indicate the $2\sigma$ confidence intervals obtained using a zero-mean prior function and a $\Lambda$CDM mean prior function, respectively, using the full marginalized posteriors, while the dashed and dotted lines show the 1$\sigma$ and 2$\sigma$ confidence intervals when optimized hyperparameters are employed. The distribution of the mean function for the corresponding hyperparameters is shown in magenta. Bottom panel shows residuals with respect to the Planck$\Lambda$CDM best-fit model. See Secs. \ref{['sec:reconstruction_comparison']} and \ref{['sec:diss_mu']} for details.
  • Figure 2: Reconstructed evolution of the Hubble parameter $H(z)$ using Gen GP (top panel). The black dotted curve denotes the Planck best-fit $\Lambda$CDM model. The blue and red shaded regions indicate the $2\sigma$ confidence intervals obtained using a zero-mean prior function and a $\Lambda$CDM mean prior function, respectively, using the full marginalized posteriors, while the dashed and dotted lines show the 1$\sigma$ and 2$\sigma$ confidence intervals when optimized hyperparameters are employed. The distribution of the mean function for the corresponding hyperparameters is also shown in magenta. Bottom panel shows residuals with respect to the Planck$\Lambda$CDM best-fit model. See Secs. \ref{['sec:reconstruction_comparison']} and \ref{['sec:diss_mu']} for details.
  • Figure 3: The left panel shows the variation of uncertainty in the Hubble constant parameter for both GP and GenGP reconstructions, using both zero mean and $\Lambda$CDM mean functions. For comparison, we also include the case where only the mean function (assumed as $\Lambda$CDM) is used, without the full GP or GenGP modeling, prior to applying the covariance matrix (Eq. \ref{['eq:meanfn']}). The right panel presents a comparison of the corresponding $\chi^2$ values for each case.