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arxiv: 2606.27252 · v1 · pith:CEGSYNYKnew · submitted 2026-06-25 · 🌌 astro-ph.EP · astro-ph.IM

Improving exoplanet mass characterisation with Bayesian model selection using the Learned Harmonic Mean Estimator

Ross S. Dobson , Vincent Van Eylen , Ioanna Manolopoulou , Kendall Sullivan This is my paper

Pith reviewed 2026-06-26 02:21 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IM
keywords exoplanetsradial velocityBayesian evidencemodel selectionMCMCplanetary massesGaussian processesvelocity trends
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The pith

The learned harmonic mean estimator allows direct estimation of Bayesian evidence from MCMC samples for radial velocity model selection in exoplanet studies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that the learned harmonic mean estimator (LHME) can compute the Bayesian evidence needed for model comparison using only standard Markov Chain Monte Carlo posterior samples from radial velocity fits. This avoids the need for separate nested sampling runs, reducing computational cost while incorporating prior information and Occam's razor that information criteria lack. The authors apply LHME to compare 18 model variants across six single-planet systems and 72 variants for a multi-planet comparison, finding that preferred models vary by system. This variation means that choosing the right model for eccentricity, noise, trends, and planet number is necessary to obtain reliable planetary masses. By integrating with existing MCMC workflows via the harmonic package, the method makes rigorous Bayesian selection practical for routine exoplanet characterization.

Core claim

The learned harmonic mean estimator estimates the Bayesian evidence directly from MCMC posterior samples with lower computational cost and no changes to the fitting procedure, enabling Bayes factor comparisons among radial velocity models that include choices for orbital eccentricity, noise models such as Gaussian processes, velocity trends, and the number of planets.

What carries the argument

The learned harmonic mean estimator (LHME), which calculates the marginal likelihood from posterior samples generated by MCMC.

If this is right

  • No single model is preferred across all systems, so model comparison is required for each exoplanet to ensure robust mass estimates.
  • Planetary masses derived from radial velocity data depend on the selected model for orbit shape, noise, and trends.
  • The method integrates with existing MCMC-based RV analysis pipelines without requiring new sampling techniques.
  • The open-source harmonic package implements LHME for broader use in astrophysics model comparisons.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If LHME evidence estimates match nested sampling on additional test cases, it could become a standard tool replacing information criteria in RV studies.
  • Re-analysis of previously published exoplanet masses with full model comparison might reveal systematic biases in some systems.
  • This approach could extend to other astronomical datasets where MCMC is already used but evidence calculation was previously expensive.

Load-bearing premise

The learned harmonic mean estimator produces accurate and stable evidence estimates for the radial velocity model classes tested, including those with Gaussian processes and multi-planet comparisons.

What would settle it

Running nested sampling on the same RV datasets and finding that the resulting evidence values differ substantially from the LHME estimates would falsify the reliability of the estimator for these applications.

Figures

Figures reproduced from arXiv: 2606.27252 by Ioanna Manolopoulou, Kendall Sullivan, Ross S. Dobson, Vincent Van Eylen.

Figure 1
Figure 1. Figure 1: An example of a harmonic diagnostic corner plot, showing the MCMC posterior samples distributions (red) for each fitted parameter, and the LHME concentrated flow (blue). The flow successfully learns the shape of the posterior distribution while the tails remain contained within it, satisfying the requirement for reliable evidence estimation with the learned harmonic mean estimator (see Section 2.3.2). The … view at source ↗
Figure 2
Figure 2. Figure 2: The derived minimum mass 𝑀p sin 𝑖 values and uncertainties for all of the valid models in the six single-planet systems. The orange vertical lines show the derived minimum mass from the literature. Models that failed harmonic diagnostics (see Section 4.4) are excluded, shown by the grey hatched areas. The row for the best-scoring model is highlighted in pink. The blue colourbar scale shows the log Bayes fa… view at source ↗
Figure 3
Figure 3. Figure 3: The derived minimum mass 𝑀p sin 𝑖 values and uncertainties for all of the valid models for TOI-544, shown for planet b (left panel) and planet c (right panel). The orange vertical lines shows the derived minimum masses from Osborne et al. (2024). Both the 18 one-planet and 54 two-planet models are shown. Models that failed harmonic diagnostics (see Section 4.4) are excluded, shown by the grey hatched areas… view at source ↗
Figure 4
Figure 4. Figure 4: Radial velocities for HD 18599 (TIC 207141131), with orange circles for HARPS03 and blue circles for HARPS15. Error bars show observed uncertainties, with lighter extensions indicating the contribution from the 𝜎jit jitter term added in quadrature. The no-trend HG0 model (upper panels), linear trend HG1 model (middle panels) and quadratic trend HG2 model (lower panels) are shown as the black curve and grey… view at source ↗
Figure 5
Figure 5. Figure 5: Phase-folded radial velocities for HD 18599 (TIC 207141131), comparing the best-scoring white-noise model HW1 (panel a) with the overall best￾scoring, preferred model HG1 (panel b), which uses a Gaussian process. In both panels, orange circles show HARPS03 and blue circles HARPS15; error bars show the observed uncertainties, with lighter extensions indicating the contribution from the 𝜎jit jitter term adde… view at source ↗
Figure 6
Figure 6. Figure 6: Eccentricity posterior samples for the half-normal prior HG0 (upper panel, blue) and uniform prior (lower panel, purple) UG0 models, for K2-265 (TIC 146364192). The prior distributions are shown with the orange lines. The half-normal prior model HG0 has a posterior peaked at 𝑒 ≈ 0.12, away from zero. The uniform prior model UG0 (purple) is peaked at 𝑒 ≈ 0.15, but shows a significant large secondary peak ne… view at source ↗
Figure 7
Figure 7. Figure 7: Phase-folded radial velocities for K2-265 (TIC 146364192). Same as [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Radial velocities for TOI-1055 (TIC 320004517), with green circles for HARPS03, orange circles for HARPS15 and blue circles for HARPS20. Error bars show observed uncertainties, with lighter extensions indicating the contribution from the 𝜎jit jitter terms added in quadrature. The white noise CW0 (upper panels) and GP CG0 (lower panels) models are shown as the black curve and grey shaded region, representin… view at source ↗
Figure 9
Figure 9. Figure 9: Phase-folded radial velocities for TOI-1055 (TIC 320004517). Same as [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Phase-folded radial velocities for TOI-220 (TIC 150098860). Same as [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: Phase-folded radial velocities for GJ 1214 (TIC 467929202). Same as [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Phase-folded radial velocities for TOI-544, shown for planet b (upper panel) and planet c (lower panel) respectively. Same as [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
read the original abstract

Radial velocity (RV) analyses require modelling choices (such as eccentricity treatment, noise model, velocity trends, and number of planets) that can significantly affect derived planetary masses. Current practice often relies on information criteria to compare and select models, but these have known limitations: they lack the built-in Occam's razor of Bayesian model comparison, and they do not incorporate prior information. Computing the Bayesian evidence needed for Bayes factor model comparison has traditionally required dedicated algorithms such as nested sampling. The learned harmonic mean estimator (LHME) offers an alternative, estimating the Bayesian evidence directly from MCMC posterior samples, with less computational cost and with no modification to the fitting procedure. We present the first application of the LHME to RV model selection, fitting 18 model variants -- comparing circular and eccentric orbits, white noise and Gaussian Process noise models, and long-term velocity trends -- to six single-planet systems, and 72 variants to a seventh system for an $N$ versus $N+1$ planet model comparison. We find that no single model is universally preferred, reinforcing the need for model comparison to select the most appropriate model for a system, thereby ensuring robust mass characterisation. The LHME, implemented in the open-source harmonic package, makes rigorous Bayesian model comparison accessible to existing MCMC-based RV workflows, and we encourage its wider use for other model comparisons in astrophysics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims that the Learned Harmonic Mean Estimator (LHME) enables computation of Bayesian evidence directly from existing MCMC posterior samples for radial velocity model selection, without modifying the fitting procedure or incurring the cost of nested sampling. It applies the method to compare 18 model variants (circular vs. eccentric orbits, white noise vs. Gaussian process noise, with/without long-term trends) across six single-planet systems and 72 variants for a seventh system to perform N vs. N+1 planet comparisons. The authors report that no single model is universally preferred and conclude that model comparison is required for robust mass characterisation, with the open-source harmonic package making the approach accessible.

Significance. If the LHME evidence estimates prove accurate and stable for these RV model classes, the work would be significant by lowering the barrier to proper Bayesian model comparison in standard exoplanet RV analyses. This could improve the reliability of derived planetary masses by incorporating Occam's razor and prior information when choosing among eccentricity treatments, noise models, and planet counts. The release of the harmonic package is a concrete strength for reproducibility and wider adoption.

major comments (1)
  1. [Abstract] Abstract and the description of the application to the seven systems: the central claim that LHME yields sufficiently accurate and stable log-evidence values for reliable model ranking (including GP kernels and multi-planet spaces) is not supported by any direct numerical comparison to nested sampling on the same likelihoods/priors, nor by tests on simulated RV datasets with independently computable true evidence. This validation is load-bearing for the assertion that the method enables 'reliable Bayesian model selection'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback. The major comment correctly identifies a gap in validation that we will address through revision.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the description of the application to the seven systems: the central claim that LHME yields sufficiently accurate and stable log-evidence values for reliable model ranking (including GP kernels and multi-planet spaces) is not supported by any direct numerical comparison to nested sampling on the same likelihoods/priors, nor by tests on simulated RV datasets with independently computable true evidence. This validation is load-bearing for the assertion that the method enables 'reliable Bayesian model selection'.

    Authors: We agree that the manuscript does not contain direct numerical comparisons of LHME log-evidence estimates to nested sampling on the same RV likelihoods/priors, nor tests on simulated datasets with known true evidence. The paper's focus is the first application of LHME to real RV model selection across the specified variants; validation of the underlying LHME estimator appears in its original references but is not repeated here for these specific model classes (GP kernels, multi-planet spaces). To strengthen the central claim, we will revise the manuscript to add: (i) a comparison of LHME versus nested-sampling evidences for a representative subset of the single-planet systems (white-noise and GP cases) using identical priors and likelihoods, and (ii) a limited test on simulated RV data for at least one simple model where the true evidence can be computed independently. These additions will be placed in a new methods/validation subsection and will directly support the reliability of the reported model rankings. revision: yes

Circularity Check

0 steps flagged

No circularity: LHME applied as independent estimator to RV models without reduction to inputs by construction

full rationale

The manuscript applies the learned harmonic mean estimator (LHME) to compute Bayesian evidence from existing MCMC posterior samples for RV model variants (circular vs eccentric, white vs GP noise, trends, N vs N+1 planets). The LHME is presented as a pre-existing method implemented in the harmonic package, with no derivation, fitting of parameters to the target evidence values, or self-referential definitions within the paper. Model selection proceeds by direct comparison of the resulting log-evidence estimates; no step renames a fitted quantity as a prediction or imports uniqueness via author self-citation chains. The central claim (that LHME enables accessible Bayesian model comparison for these RV analyses) rests on the estimator's properties as an external tool rather than on any internal reduction. Absence of controlled simulations or nested-sampling benchmarks is a verification gap, not a circularity issue.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on the accuracy of LHME evidence estimates for RV models (assumed from prior LHME literature) and on the premise that information criteria are inferior for this task.

axioms (1)
  • domain assumption Bayesian evidence comparison via Bayes factors is preferable to information criteria for model selection in RV analyses
    Abstract asserts that information criteria lack built-in Occam's razor and prior incorporation.

pith-pipeline@v0.9.1-grok · 5789 in / 1055 out tokens · 37567 ms · 2026-06-26T02:21:28.807836+00:00 · methodology

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