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

This paper demonstrates the first application of the Learned Harmonic Mean Estimator (LHME) to radial velocity exoplanet analyses, showing that this computationally efficient method enables rigorous Bayesian model selection across various orbital and noise models to ensure robust mass characterisation without requiring dedicated nested sampling algorithms.

Original authors: Ross S. Dobson, Vincent Van Eylen, Ioanna Manolopoulou, Kendall Sullivan

Published 2026-06-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ross S. Dobson, Vincent Van Eylen, Ioanna Manolopoulou, Kendall Sullivan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to figure out the weight of a ghost that is tugging on a star. You can't see the ghost (the exoplanet), but you can see the star wobble back and forth. The size of that wobble tells you how heavy the ghost is.

This paper is about making sure you get the weight right, even when the data is messy, the ghost is hiding, or there might be two ghosts instead of one.

Here is the story of how the authors solved this problem, explained simply:

The Problem: Too Many Ways to Guess

When astronomers look at these wobbles, they have to make a lot of guesses about how the data works. It's like trying to solve a puzzle where you aren't sure if the pieces are square or round, if the picture is blurry or sharp, or if there's a smudge on the lens.

The authors had to decide:

Is the planet's orbit a perfect circle, or is it a squashed oval?
Is the "noise" in the data just random static, or is it caused by the star itself acting up (like sunspots)?
Is the star drifting slowly over time because of a hidden, distant companion?

In the past, scientists used simple rules of thumb (like counting how many variables they used) to pick the best guess. But the authors say these rules are like using a ruler to measure the temperature of a soup—they just don't work well enough. They often miss the subtle clues that tell you which model is actually true.

The Solution: The "Learned Harmonic Mean Estimator" (LHME)

The authors introduced a new tool called the Learned Harmonic Mean Estimator (LHME).

Think of the old way of doing this as trying to calculate the average height of a crowd by asking everyone to shout out their height, but some people are whispering and others are screaming. It's messy and hard to get a true average.

The new tool (LHME) is like a smart AI translator. It takes all the messy, shouting data the astronomers already collected and "learns" the true shape of the crowd. It doesn't need to restart the whole investigation; it just looks at the notes the astronomers already wrote down (the computer simulations they ran) and figures out the most probable answer.

It's special because:

It's fast: It doesn't require a supercomputer to run a new, different type of test. It works with the data you already have.
It's fair: It naturally penalizes complicated guesses that aren't supported by the evidence. If you guess there are two ghosts, but the data only clearly shows one, this tool says, "Nope, that's too complicated for what we see."

What They Did

The team tested this new tool on seven different star systems.

For six of them, they had one known planet. They tried 18 different versions of the math for each star (changing the orbit shape, the noise model, and the drift).
For the seventh star (TOI-544), they weren't sure if there was one planet or two. They tested 72 different versions to see if the data supported a second, invisible planet.

What They Found

The results were surprising and important:

There is no "One Size Fits All": The best way to measure the weight of a planet on Star A is different from the best way for Star B. You can't just pick one rule for the whole universe.
Circular vs. Oval: For most of the single planets, the data suggested they were moving in perfect circles. The "oval" guesses were usually too complicated for the data to support.
The "Uniform" Trap: They found that using a "uniform" guess for the orbit shape (where every shape is equally likely) often led to crazy, wrong answers. Using a "half-normal" guess (which assumes orbits are usually circular but can be oval) worked much better.
The Second Ghost: For the star TOI-544, the new tool confirmed with high confidence that there is a second planet, even though we can't see it. The math strongly preferred the "two-planet" model over the "one-planet" model.
Mass Matters: The biggest takeaway is that the choice of math changes the weight. In some cases, picking the wrong model made the planet look three times heavier than it actually is. Since the weight tells us if a planet is a rocky rock or a gas ball, getting this wrong changes our whole understanding of what the planet is made of.

The Bottom Line

This paper shows that to get the right answer about how heavy an alien world is, you have to try many different mathematical "outfits" and use a smart, fair judge (the LHME) to pick the winner. It's not enough to just pick the first outfit that looks okay; you have to check them all to make sure you aren't fooling yourself.

The authors have made their tools (called ravest and harmonic) available for free so other astronomers can use this "smart translator" to get better answers about the universe.

Problem
Radial velocity (RV) analyses for exoplanet mass characterisation rely on numerous modelling choices, including the treatment of orbital eccentricity, the noise model (white noise vs. Gaussian Processes), the inclusion of long-term velocity trends, and the number of planets in the system. These choices significantly impact derived planetary masses, densities, and inferred formation pathways. Current practices often rely on information criteria (AIC, BIC) or goodness-of-fit statistics ( $\chi^2$ , $\ln \mathcal{L}$ ) for model selection. However, these metrics lack the built-in "Occam's razor" of Bayesian model comparison, do not incorporate prior information, and treat all free parameters equally regardless of how well the data constrains them. While Bayesian model comparison via Bayes factors is the rigorous standard, it requires calculating the Bayesian evidence (marginal likelihood), a high-dimensional integral traditionally requiring computationally expensive dedicated algorithms like nested sampling, which often necessitates replacing standard MCMC fitting workflows.

Methodology
The authors apply the Learned Harmonic Mean Estimator (LHME), implemented in the open-source harmonic Python package, to RV model selection. The LHME estimates the Bayesian evidence directly from posterior samples generated by standard MCMC methods (specifically using the emcee sampler via the ravest package), decoupling evidence estimation from the sampling strategy. This allows for post-hoc model comparison using existing MCMC chains without modifying the fitting procedure.

The study analyses seven systems:

Six single-planet systems (K2-265, TOI-220, HD 18599, LHS 1815, TOI-1055, GJ 1214) drawn from a homogeneous re-analysis of archival HARPS data. For each, 18 model variants were fitted, varying:
- Eccentricity: Circular ( $e=0$ ), Uniform priors on $\sqrt{e}\cos\omega_*$ and $\sqrt{e}\sin\omega_*$ , or Half-normal priors on $e$ .
- Noise: White noise with jitter vs. Gaussian Process (GP) with a quasi-periodic kernel.
- Trends: None, linear, or quadratic velocity trends.
TOI-544: A two-planet system used to demonstrate $N$ vs. $N+1$ planet model comparison. Here, 72 variants were fitted (18 one-planet and 54 two-planet configurations).

The authors employ strict diagnostic checks (kurtosis, variance stability, and visual inspection of corner plots) to exclude models where the normalising flows in harmonic fail to approximate the posterior (e.g., due to multimodality often caused by uniform eccentricity priors fitting spurious high-e solutions).

Key Contributions

First Application to RV: This is the first application of the LHME to RV model selection.
Workflow Integration: It demonstrates that rigorous Bayesian evidence estimation can be integrated into existing MCMC-based RV workflows with negligible additional computational cost and no modification to the fitting procedure.
Systematic Comparison: The paper provides a systematic comparison of 18–72 model variants across seven diverse systems, moving beyond single-model assumptions to quantitatively determine the most appropriate model for each system.

Results

No Universal Model: No single model combination (eccentricity, noise, trend) is universally preferred. The optimal model depends on the specific system.
Eccentricity Priors: Uniform eccentricity priors consistently underperformed, often failing diagnostic tests due to multimodal posteriors with spurious high-eccentricity solutions. Half-normal priors (informative) were consistently preferred or produced better-behaved posteriors.
Noise Models: GP noise models were preferred for systems with independent evidence of rotationally modulated stellar activity (K2-265, HD 18599, TOI-1055, GJ 1214, TOI-544). White noise models were preferred for systems where activity was not dominant (TOI-220, LHS 1815).
Trends: Long-term trends were only preferred in two systems (HD 18599 with a linear trend attributed to a known companion; TOI-220 with a quadratic trend).
$N$ vs. $N+1$ Planets: For TOI-544, the Bayesian evidence strongly favored the two-planet model over the one-planet model ( $\Delta \ln Z = 20.1$ ), confirming the detection of the non-transiting companion.
Impact on Mass: Model choice significantly affects derived masses. In some cases (e.g., TOI-1055), the choice between white noise and GP models resulted in a factor of three difference in the semi-amplitude and derived mass. In others (e.g., GJ 1214), masses were robust across models.

Significance and Claims
The paper claims that the LHME makes rigorous Bayesian model comparison accessible to the RV community, overcoming the computational and workflow barriers of nested sampling. The authors emphasize that relying on information criteria (AIC/BIC) or single-model assumptions can lead to incorrect mass characterisations, as these metrics do not penalise complexity in a way that accounts for prior information or data constraints. By demonstrating that different systems require different modelling choices, the authors argue that fitting multiple variants and comparing them via Bayesian evidence is essential for robust mass characterisation, which in turn is critical for understanding exoplanet compositions, formation pathways, and habitability. The work encourages the wider use of the LHME for other model comparisons in astrophysics where MCMC posterior samples exist.

The Problem: Too Many Ways to Guess

The Solution: The "Learned Harmonic Mean Estimator" (LHME)

What They Did

What They Found

The Bottom Line

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