HERMES: HiERarchical Modelling for Exoplanet Science

Abstract: ESA's Ariel Space Mission will characterise the atmospheres of approximately 1000 exoplanets to quantify population-level trends. We present HERMES (HiERarchical Modelling for Exoplanet Science), a multidimensional Bayesian framework that probes population-level correlations across multiple axes of diversity. The specific use case we present is the multidimensional relation between stellar metallicity, planetary mass, and atmospheric metallicity. Starting from the Ariel Mission Candidate Sample (Edwards & Tinetti 2022), we select confirmed planets with available masses and stellar metallicity, inject plausible multidimensional trends and demonstrate successful parameter recovery. Simulated surveys are generated with a variety of leverage and sample size, in the presence of intrinsic astrophysical scatter and measurement noise. By fitting independent Bayesian models to each survey, we confirm that survey leverage remains a reliable predictor of trend precision even in multiple dimensions and in the presence of intrinsic astrophysical scatter. For an Ariel Tier 2 transit survey of at least 400 planets, HERMES robustly recovers the correlation between stellar and planetary metallicity despite intrinsic scatter in planetary abundances as large as 1.2 dex. These results establish HERMES as a practical tool for survey design and science yield forecasting in preparation for Ariel and other surveys probing population-level trends.

• The paper presents a hierarchical Bayesian model that recovers injected coefficients for planetary mass and host-star metallicity despite measurement errors and intrinsic scatter.
• It demonstrates that slope uncertainty scales inversely with mass leverage, highlighting the need to balance sample size and leverage in multidimensional survey designs.
• WAIC-based model comparisons confirm that including host-star metallicity as a predictor significantly enhances the detection of exoplanet atmospheric trends in large surveys.

HERMES: Hierarchical Bayesian Modelling for Population-Level Exoplanet Atmospheres

Introduction

Large-scale comparative studies of exoplanet atmospheres require robust statistical frameworks to extract population-level correlations from heterogeneous, noisy, and intrinsically diverse datasets. The ESA Ariel mission, targeting spectroscopic characterization of approximately 1000 exoplanets, necessitates inference tools capable of quantifying multidimensional trends—most notably, the interplay among planetary mass, host-star metallicity, and planetary atmospheric metallicity. The HERMES framework ("HiERarchical Modelling for Exoplanet Science") is a multidimensional Bayesian hierarchical model specifically designed to meet these requirements. HERMES generalizes population-level trend retrieval to multidimensional spaces, rigorously accounting for measurement errors, intrinsic astrophysical scatter, and survey design considerations (2606.02696).

Theoretical Framework and Model Structure

HERMES implements a hierarchical Bayesian linear regression that simultaneously models planetary atmospheric metallicity (in log space, using water abundance as a proxy) as a function of both planetary mass and host-star metallicity. Key methodological aspects include:

• Covariate Uncertainty Propagation: Host-star metallicities are treated as latent variables to prevent errors-in-variables bias. Measurement uncertainty is explicitly propagated, following the recommendations of [Kelly 2007].
• Intrinsic Astrophysical Scatter: The intrinsic scatter parameter $\varepsilon$ models planet-to-planet diversity beyond measurement uncertainties, capturing stochasticity in planet formation, migration, atmospheric loss, and unmodeled covariates.
• Survey Leverage: The concept of leverage, defined as the normalized quadrature sum of covariate deviation, is extended to multidimensional survey designs, generalizing the single-axis approach presented by Cowan et al. (2025).
• Model Comparison: Two models are systematically compared: a 2D model (planetary mass only) and the full 3D model (mass and stellar metallicity). The distinction between intrinsic scatter and a true dependence on stellar metallicity is assessed using the Watanabe–Akaike Information Criterion (WAIC), suitable for hierarchical Bayesian models.

The model is implemented via NUTS Hamiltonian Monte Carlo sampling using NumPyro/JAX. Priors are empirically calibrated using the observed sample mean and variance.

Simulated Survey Construction

Mock surveys are constructed from the Ariel Mission Candidate Sample (MCS), restricted to planets with measured masses and stellar metallicities, yielding 858 systems. Mass leverage is manipulated via systematic sample truncation: four nested mass classes (S1 through S4) span decreasing mass ranges. Multiple survey sizes are tested, with comprehensive coverage of both sample size ($N$) and leverage across the covariate axes.

Artificial population-level trends are injected by specifying predetermined coefficients for mass and stellar metallicity dependence (following physical expectations from previous exoplanet and Solar System studies), with controlled additions of Gaussian intrinsic scatter.

Numerical Results and Robustness

Strong numerical results reported include:

• Parameter Recovery: Across 180 mock surveys differing in sample size and leverage, the hierarchical Bayesian posterior robustly recovers the injected trend coefficients for both mass and stellar metallicity axes. Calibrated $z$-score statistics for all parameters ($\alpha_p$, $\beta_p$, $\beta_s$, $\varepsilon$) are consistent with well-behaved posteriors (standard deviations ≈1, coverage fractions within Gaussian expectations).
• Scaling of Precision: The uncertainty in the mass–metallicity slope $\beta_p$ scales tightly with the mass leverage $L_p$, approximately as $\sigma_{\beta_p} \propto L_p^{-1}$, regardless of sample size. Increasing $N$0 improves precision for all parameters, but for slope recovery, leverage along the covariate is a more efficient control parameter than $N$1 alone.
• Disentangling Stellar Effects: Recovery of the stellar metallicity effect ($N$2) is sensitive to both the range and uncertainty of stellar metallicities in the sample. Even for sizable astrophysical scatter ($N$3 dex), Ariel-scale samples ($N$4) consistently distinguish the stellar–planetary metallicity correlation from intrinsic scatter, whereas smaller surveys lose sensitivity.
• Model Selection: WAIC-based model comparison demonstrates that for large samples, the inclusion of stellar metallicity as a predictor is statistically justified even in the presence of substantial intrinsic scatter, with the 3D model favored in nearly all trials for $N$5.

Implications for Exoplanet Population Inference

The HERMES framework has several important ramifications for exoplanet atmospheric population surveys:

• Survey Optimization: Both leverage and sample size must be jointly optimized to maximize information about multidimensional population trends. Maximizing leverage along one axis may come at the expense of other independent axes if not properly balanced. Techniques such as simulated annealing can identify survey designs that balance these trade-offs (Panek et al., 28 Jan 2026).
• Astrophysical Scatter and Survey Scale: The practical detection threshold for secondary (e.g., stellar metallicity) dependencies is set jointly by sample size and intrinsic scatter. For highly stochastic populations, only the largest surveys retain the statistical power to disentangle subtle population-level correlations.
• Robustness to Measurement Uncertainty: Propagating covariate uncertainties prevents bias and ensures interpretability of recovered population parameters. This is essential for the reliable inference of exoplanet atmospheric formation processes.
• Generality of Approach: HERMES is not limited to atmospheric metallicity; it is extendable to any response variable expected to depend on multiple axes of planetary or stellar diversity, and can be applied both to survey design (forecasting science yield) and to analysis of real observational data.

Theoretical and Practical Implications

Theoretically, HERMES demonstrates that multidimensional hierarchical Bayesian inference for exoplanet populations can achieve well-calibrated, interpretable posteriors even in the presence of large intrinsic scatter and substantial measurement uncertainty. The explicit diagnostics for intrinsic scatter provide a falsifiable pathway for model refinement: unexplained variance can be systematically traced to missing physical predictors.

Practically, this suggests that robust detection of population-level exoplanet trends—particularly for Ariel and similar ambitious survey missions—is achievable, provided that survey size and diversity are sufficient. HERMES quantifies the observational requirements (in $N$6 and in leverage) necessary for constraining subtle but physically significant effects such as planet–star composition coupling.

Future developments may include applying HERMES to higher-dimensional population models, integrating additional axes of diversity such as planetary irradiation, host-star age, or elemental abundance ratios (e.g., C/O), and deploying more flexible non-linear or nonparametric structures.

Conclusion

HERMES establishes a rigorous, multidimensional hierarchical Bayesian modeling framework for exoplanet population studies, with explicit accommodation of measurement uncertainties, sample size effects, astrophysical scatter, and the multidimensional optimization of survey design (2606.02696). Numerical simulations empirically validate its recovery of injected trends and its discrimination between true population correlations and intrinsic diversity. The results provide a statistically principled foundation for Ariel-era exoplanet science, ensuring that atmospheric population-level inferences will be robust to the complexities of astrophysical and observational noise, and guiding the practical design and scientific prioritization of future exoplanet surveys.