Comparative Analysis of EMCEE, Gaussian Process, and Masked Autoregressive Flow in Constraining the Hubble Constant Using Cosmic Chronometers Dataset

1. Problem Statement

The Hubble constant ($H_0$) is a fundamental parameter in cosmology, yet a significant discrepancy (the "Hubble tension") exists between early-universe measurements (Planck CMB: $67.4 \pm 0.5$ km s$^{-1}$ Mpc$^{-1}$) and late-universe measurements (Cepheid-calibrated Supernovae: $73.04 \pm 1.04$ km s$^{-1}$ Mpc$^{-1}$).

Cosmic Chronometers (CC) offer a model-independent method to measure the Hubble parameter $H(z)$ at various redshifts, providing an alternative route to constrain $H_0$. However, different statistical inference methods applied to the same CC dataset yield varying $H_0$ values. The authors aim to systematically compare three prominent methods—Affine Invariant Markov chain Monte Carlo Ensemble sampler (EMCEE), Gaussian Process (GP), and Masked Autoregressive Flow (MAF)—to determine:

1. How sensitive each method is to individual data points.
2. Which method provides the most accurate and robust constraints on $H_0$ under controlled conditions.

2. Methodology

Data Source

The study utilizes the Cosmic Chronometers (CC) dataset, comprising $N=33$ independent $H(z)$ measurements spanning redshifts $z \in [0.07, 1.965]$. The data is treated as independent with published uncertainties.

Inference Methods

The authors apply three distinct approaches to constrain $H_0$:

1. EMCEE: A parametric Bayesian approach assuming a flat $\Lambda$CDM cosmological model. It samples the posterior $P(H_0, \Omega_M | H_{obs})$ using a likelihood function based on Gaussian errors.
2. Gaussian Process (GP): A non-parametric, model-independent reconstruction of $H(z)$. It uses a Squared Exponential covariance function to reconstruct the function and extrapolates to $z=0$ to derive $H_0$.
3. MAF (Masked Autoregressive Flow): A deep learning-based normalizing flow. It is trained on simulated parameter-data pairs (generated from $\Lambda$CDM) to learn the conditional posterior density $P(\theta | H_{CC})$. Once trained, it infers $H_0$ from the real CC data.

Sensitivity Analysis: Monte Carlo Delete-$d$ Jackknife (MCDJ)

To assess sensitivity to individual data points, the authors employ the MCDJ technique:

• Procedure: From the 33-point dataset, they randomly select 26 points (deleting 7, or $d=7$) and repeat this process 1,000 times.
• Metric: They analyze the distribution of the resulting 1,000 $H_0$ posterior central values for each method.
• Redshift Dependence: The dataset is split into low-redshift ($z < 0.48$) and high-redshift ($z \geq 0.48$) subsets to test if sensitivity varies with redshift.

Simulation-Based Performance Validation

To evaluate accuracy and calibration, the authors generate two ensembles of 100 mock CC datasets with a known "ground truth" $H_{0,true}$:

1. $\Lambda$CDM-based Simulation: Mock data generated using the flat $\Lambda$CDM model with $H_{0,true} = 67.75$.
2. GP-based Simulation: Mock data generated using a GP reconstruction of the observed data as the fiducial model with $H_{0,true} = 67.21$.

Performance Metrics:

• Accuracy: Bias (systematic offset) and Root-Mean-Square Error (RMSE).
• Calibration: Empirical coverage of 68% and 95% credible intervals (Cov68, Cov95).
• Posterior Quality: Log Score (log posterior density at the true value).

3. Key Contributions

1. Systematic Sensitivity Quantification: The paper introduces a rigorous MCDJ framework to quantify how much individual $H(z)$ data points influence the final $H_0$ constraint for different inference pipelines.
2. Redshift-Dependent Sensitivity Discovery: It reveals that sensitivity to data points is not uniform across methods or redshifts.
3. Comprehensive Benchmarking: It provides a "like-for-like" comparison of parametric (EMCEE), non-parametric (GP), and machine learning (MAF) methods on the same dataset under identical conditions, moving beyond previous studies that used different datasets or implementations.
4. Bias Identification in GP: The study identifies a persistent negative bias in GP-derived $H_0$ values that cannot be attributed solely to systematic shifts in the input mock data, suggesting intrinsic limitations in the GP reconstruction for this specific application.

4. Results

Sensitivity to Individual Data Points (MCDJ)

• Overall Sensitivity: GP is the most sensitive to the removal of individual data points, followed by MAF, with EMCEE being the least sensitive.
  • Evidence: The spread ($\Delta H_0$) of the 1,000 posterior central values is largest for GP ($\Delta H_0, 2\sigma \approx 11.77$) and smallest for EMCEE ($\approx 5.18$).
• Redshift Dependence:
  • EMCEE & GP: More sensitive to high-redshift ($z \geq 0.48$) data points.
  • MAF: More sensitive to low-redshift ($z < 0.48$) data points.
  • Interpretation: EMCEE and GP rely on the global shape of the expansion history (high-$z$ leverage), while MAF, trained on local normalization, anchors more heavily on low-$z$ data.

Simulation-Based Performance

Across both $\Lambda$CDM and GP-based simulations, the ranking of methods is consistent:

Metric	EMCEE	GP	MAF
Bias	Small (Mildly negative)	Largest Negative Bias	Negligible (but high variance)
RMSE	Lowest (Most accurate)	Intermediate	Highest (Least stable)
Coverage (68%)	Conservative (Over-coverage)	Near Nominal	Under-coverage
Coverage (95%)	Conservative	Near Nominal	Under-coverage (Over-confident)
Log Score	Highest (Best quality)	Intermediate	Lowest

• EMCEE: Delivers the tightest credible intervals, lowest RMSE, and highest log score. It is the most robust against data perturbations.
• GP: Shows the largest systematic negative bias in $H_0$ (approx. -2 to -3 km s$^{-1}$ Mpc$^{-1}$) and high sensitivity to data deletion.
• MAF: While unbiased on average, it suffers from the largest RMSE (high realization-to-realization variability) and the lowest log score, indicating poor posterior quality.

5. Significance and Conclusions

• Recommendation for $H_0$ Inference:

  • EMCEE is recommended as the default method for CC-only $H_0$ constraints if a parametric model ($\Lambda$CDM) is acceptable. It offers the best balance of accuracy, stability, and calibration.
  • GP is valuable as a model-independent diagnostic and for reconstructing the $H(z)$ function shape, but it should not be used as the primary precision estimator for $H_0$ due to its high sensitivity to data points and systematic bias.
  • MAF is not recommended as the default for this specific dataset. Despite its speed after training, it fails to outperform traditional methods in accuracy or stability on current CC data.

• Implications for the Hubble Tension: The study highlights that the choice of inference pipeline significantly impacts the derived $H_0$ value. To resolve tensions or compare results across studies, researchers must report not just the central value, but also robustness diagnostics (like MCDJ) and simulation-based validation metrics (bias, coverage, log score).

• Future Work: The authors note that the limited size of the current CC dataset (33 points) may amplify these sensitivity differences. Future studies should test these methods on larger datasets and investigate the impact of correlated systematics, which are currently assumed to be negligible.