Going Bayesian on the ages of nearby young stellar systems I. The expansion rate method

A. Berihuete; H. Bouy; J. Olivares

REVIEW 3 major objections 6 minor 44 references

Going Bayesian on the ages of nearby young stellar systems I. The expansion rate method

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A fully Bayesian expansion-rate estimator recovers ages of nearby young stellar associations within 10 percent, with 100 percent credibility on simulated systems.

desk verdict Useful Bayesian wrapper for expansion ages; the validation is real but the abstract sells it as method performance when the numbers are conditional on truth-centered priors. read the letter →

arxiv 2506.05110 v1 pith:RXLDSIXA submitted 2025-06-05 astro-ph.SR astro-ph.GAastro-ph.IM

J. Olivares , A. Berihuete , H. Bouy This is my paper

classification astro-ph.SRastro-ph.GAastro-ph.IM

keywords expansionratemethodBayesianhierarchicalmodelstellaragesyoungassociationsKalkayotllinearvelocityfieldcredibleintervalsstar-formingregions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper builds a Bayesian version of the expansion-rate method for dating young stellar systems, a technique that turns the measured rate at which a stellar association is spreading apart into an age. The authors' goal is to fix known weaknesses of the frequentist approach: neglected correlations between expansion components, underestimated uncertainties, and biased ages from inverting a noisy expansion rate. They implement the method as a hierarchical model in the public code Kalkayotl in which the age is a sampled parameter with a prior informed by literature age estimates, and they validate it on a large grid of simulated associations and star-forming regions patterned on Beta-Pictoris and Taurus. On simulated stellar associations with ages 10-40 Myr and distances up to 150 pc the estimator returns errors below 10 percent and 95% credible intervals that always contain the true age, outperforming the classical and robust frequentist estimators on the same data. If these results transfer to real systems, the method gives the field a dating tool whose uncertainties can be trusted and combined with other techniques.

What carries the argument

The load-bearing object is the Bayesian hierarchical age model built on the Lindegren linear velocity field, where the expansion age τ is a sampled parameter at the top of the hierarchy. From τ the model forms the expansion scale κμ = (1.02271 τ)^{-1}, draws the three Cartesian expansion components from a common Normal or Student-t distribution with scale κσ (with a Gamma prior on the Student-t degrees of freedom), and feeds them into the linear velocity field. A Generalised Gamma (left) prior with hyper-parameters set from literature age estimates embeds prior knowledge while allowing independent mode and variance, and an Exponential prior on κσ plus Gamma/Exponential priors on position and velocity dispersions regularize the inference. Sampling the full posterior with the NUTS algorithm propagates data uncertainties and correlations directly into the age posterior, avoiding the error leaks of successive propagation steps.

What would settle it

Run the method on a real stellar association whose age is independently known from a non-dynamical technique, using a literature prior that is deliberately shifted away from that age; if the 95% HDI excludes the true age, the claimed credibility does not hold outside the simulation's favorable prior conditions.

Watch

Extended reading notes

Core claim

The central claim is that the expansion rate method, reformulated as a Bayesian hierarchical model, is a more robust and credible age estimator than the frequentist implementations currently used. The age τ is treated as a random variable drawn from a Generalised Gamma (left-skewed) or Truncated Normal prior, converted into an expansion scale κμ = (1.02271 τ)^{-1}, and then the three expansion components κX, κY, κZ are drawn from a common normal or Student-t distribution with that scale; drawing them jointly propagates their correlations into the age posterior. Validated on synthetic associations (10-40 Myr, 50-150 pc, 25-75 members, 25%-100% radial-velocity coverage), the method yields errors within ±10%, uncertainties below 20%, and 100% credibility in all tested configurations, whereas classical and robust frequentist estimators degrade with age, distance, and sparse radial-velocity coverage, with credibilities dropping below 50% in the 20 and 40 Myr cases. In simulated star-forming regions the method is less accurate, with errors up to 80% at 3 Myr and 400 pc, but credibility remains 100% and the error decreases rapidly with age. The paper argues this validates the Bayesian formulation as the preferred implementation of the expansion rate method within its applicability domain.

Load-bearing premise

The validation performance depends on the age prior being centered near the true age and on informed phase-space dispersion priors set to the true values; if the literature age prior used for a real system is biased or too narrow, the Bayesian estimator inherits that bias, and the headline '<10% error, 100% credibility' result is not guaranteed.

Editorial extensions

If this is right

Users of Kalkayotl can now obtain expansion ages with fully propagated uncertainties and a built-in prior check comparing prior and posterior, rather than point estimates from separate regressions.
The 100 percent credibility on simulated associations means the 95% HDI can be read as a reliable uncertainty interval, which the frequentist estimators fail to provide even at 10 Myr when radial-velocity coverage is incomplete.
Doubling the number of members with sparse radial velocities beats halving membership with full radial velocities, so membership completeness should be prioritised in target selection.
For the nearest and youngest star-forming regions (around 3 Myr, under 150 pc), the method should be used cautiously or not at all, since errors can reach 80%.
The uncertainty estimate behaves predictably, decreasing with more members and more radial-velocity coverage, unlike the erratic uncertainties of the frequentist estimators at older ages.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next step is to combine this dynamical age with isochrone or lithium-depletion ages in a single joint hierarchical model, letting the data arbitrate the offsets between techniques; the paper notes this as its own future work.
The asymmetry found under shifted priors (a +5 Myr shift biases the result more than a -5 Myr shift) suggests that using a literature age prior that is systematically too old is more dangerous than one that is too young; users could test this by running the model with both a central and a deliberately younger prior and comparing posteriors.
If the method is applied to a real association whose literature age comes from a technique that is itself biased, the claimed credibility will not hold, so the prior-check plots are not a diagnostic ornament but the main safeguard.
Testing on real associations with independently calibrated ages would be the decisive transfer check, since the paper's validation uses priors centered at or near the true age in the simulations.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something genuinely useful: it turns the expansion-rate age estimator into a single Bayesian hierarchical model, samples the age itself from literature-inspired priors, and ships the whole thing as a module of Kalkayotl. The NUTS implementation, the grid of simulations, and the head-to-head comparison against a classical and a 'robust' frequentist estimator are careful and reproducible. The comparison is fair, and the Bayesian estimator does handle correlations and low-information datasets better than the frequentist baselines, which fall apart at 20–40 Myr.

What is new is the age-as-parameter hierarchy over the Lindegren linear velocity field, and the Student-t option for treating a non-isotropic expansion component as an outlier. That is a natural extension of the existing Kalkayotl Gaussian model, but it is still a real step forward for practice.

The soft spot is the validation framing, not the machinery. Section 5.1 validates with informed phase-space dispersion priors set to the true Table 1 values, and the age prior is centered on the true age (23 Myr for the β-Pictoris template). Section 4.2 then shows that shifting the age prior by +5 Myr gives 10–15% errors, and Fig. 11 shows that a 10 Myr prior (true age 23 Myr) yields a posterior whose 95% HDI excludes the true age. So the abstract's '<10% error' and '100% credibility' are conditional on the prior being close to right, and for a real system that is not guaranteed. To their credit, the authors do not hide this: Sect. 6.2 recommends weakly informative priors, prior-posterior checks, and shifted-prior experiments, and Appendix A is an honest list of assumptions. The mismatch is between the hedged body and the unhedged abstract/conclusions. That is the main revision I would demand: restate the headline claims with the prior-accuracy condition attached, and add a validation run with deliberately broad or shifted priors to show what users should expect when the literature age is off.

Minor points: the 'better' errors for 25-source systems are partly prior dominance, which they say in Sect. 4.3; the text has a few typos; and the runtime (hours vs. minutes for frequentist) deserves a mention in the conclusions as a practical caveat.

This is a paper for people who date nearby young associations and want credible intervals on dynamical ages. The code is a real deliverable and the statistical work is solid. It deserves a serious referee. I would recommend conditional acceptance after the abstract is rewritten and a blind-prior validation is added.

Referee Report

3 major / 6 minor

Summary. The paper presents a Bayesian hierarchical implementation of the expansion rate method for estimating the ages of nearby young stellar systems, released as a module of the open-source code Kalkayotl. The model extends an existing linear velocity field model with an age parameter whose prior encodes literature age estimates, and samples the posterior with NUTS within PyMC. The authors benchmark the method against classical and robust frequentist estimators on synthetic stellar associations and star-forming regions, reporting errors below 10% and 100% credibility for associations between 10 and 40 Myr and up to 150 pc, and larger but decreasing errors for star-forming regions.

Significance. If the performance claims held under realistic prior misspecification, this would be a valuable contribution: it provides an open-source, fully Bayesian expansion-age estimator with documented assumptions, an extensive synthetic-data validation, and a useful sensitivity analysis. The use of NUTS with explicit prior/posterior checks and the public code release are clear strengths. However, the headline validation is conditioned on priors centred on the true simulation values, which is exactly the condition under which a Bayesian method is least informative and under which the 100% credibility metric is expected. The sensitivity tests in the paper itself demonstrate prior-shift errors of 10-15% and an HDI that excludes the true age for a younger prior, so the abstract's unqualified '<10% error' claim overstates the method's robustness.

major comments (3)

[Sect. 5.1 and Sect. 3.2.2] The headline validation result is conditioned on priors that already encode the true simulation values. The age prior is set to mu_tau=23 Myr, sigma_tau=8 Myr (the true age of the beta-Pictoris template), and the phase-space dispersion priors are selected as 'informed' using the generating values in Table 1. The paper's own sensitivity analysis in Sect. 4.2 shows that a +5 Myr shift in the prior yields 10-15% errors, and Fig. 11 shows that a prior centred at 10 Myr (true age 23 Myr) produces a posterior 95% HDI of [14.2, 22.9] Myr, which excludes the true age. Therefore the abstract's unqualified claim that 'errors are <10%' is a statement about performance conditional on an accurate external age estimate, not a property of the method under the conditions in which it will be used. I recommend either recalibrating the claim (e.g., reporting errors as a function of prior offset) or adding a validation protocol that uses priors drawn from a realistic, possibly biased, distribution of literature ages.
[Sect. 4.1 and Sect. 6.1] The '100% credibility' metric is in large part produced by the prior rather than by the data. Whenever the prior is centred on the true age and its scale is comparable to the true scatter (sigma_tau=8 Myr at 23 Myr), the 95% HDI is almost guaranteed to contain the true value even with modest data; Sect. 4.3 explicitly notes that for 25-source systems the prior dominates. The comparison against frequentist estimators is therefore not apples-to-apples: the Bayesian estimator receives external age information that is centred on the answer, while the frequentist estimators do not. The claim that 'the Bayesian age estimator is more robust and credible than the commonly used frequentist ones' (abstract) is not established by this comparison; a fairer test would give the frequentist estimators access to the same external age information, or would evaluate both with no prior or with the same prior misspecification.
[Sect. 6.2 and Fig. 11] The manuscript itself states that 'the most important source of bias in our method is the a priori information embedded in the age prior' and recommends using weakly informative priors. This is a sound caveat, but it directly contradicts the unqualified abstract claim. For a real stellar system, a literature age prior can be biased or too narrow, precisely the condition under which the method's posterior can exclude the true age (Fig. 11). The authors should either move the prior-sensitivity results into the abstract and conclusions, or restrict the performance claim to cases where the prior is known to be reliable.

minor comments (6)

[Sect. 2.1] The phrase 'the signal-to-noise ration' should be 'the signal-to-noise ratio'.
[Sect. 3.2.2, Table 2] The shape parameter p of the Generalised Gamma distribution is listed in the table but not defined in the text; please define p and explain the f* factor in the caption or in a footnote.
[Sect. 3.2.1, Eq. (1)] The origin of the conversion factor 1.02271 in Eq. (1) is not explained; a sentence stating the unit conversion between km/s/pc and Myr would help.
[Sect. 4.1] The phrase 'probed to minimise' should be 'proved to minimise' or 'was found to minimise'.
[Sect. 4.3] The sentence 'This effect is due to the effect of the prior' is redundant; consider rephrasing to 'This effect is due to the prior, which is centred on the true value and dominates in low-information datasets.'
[Sect. 5.2] The phrase 'we advice caution' should be 'we advise caution'.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline '<10% errors / 100% credibility' is validated with priors centered on the true age and true phase-space dispersions; the paper's own sensitivity tests show the prior dominates in the low-information cases included in that headline, so the validation partially reduces to the prior input.

fitted input called prediction [Sect. 4.3 and Fig. 4; Abstract 'errors are <10%']
"We notice that the age error in systems with 25 sources is smaller than that of systems with 50 sources. This effect is due to the effect of the prior, which is centred in the true value and dominates under these low-information content datasets."

The validation grid includes 25-source systems, and the paper explicitly says their small errors come from the age prior being centered on the true value and dominating the posterior. The abstract presents <10% errors and 100% credibility as properties of the estimator, but in low-information datasets the posterior mean is essentially the prior mean; since the prior mean was set to the true age, the 'prediction' is statistically forced by the prior. Fig. 11 confirms this: a 10 Myr prior for a true 23 Myr system gives a 95% HDI [14.2, 22.9] Myr excluding the true age. Real literature priors are not guaranteed to be centered on the truth, so the unconditional headline is not supported.
other [Sect. 5 opening; Table 1 informed phase-space dispersion values]
"In this section, we present the validation of our new Bayesian expansion rate method (Sect. 3) with its default configuration of prior distributions and hyper-parameters values (see Sect. 4; with the informed hyper-parameters for phase-space dispersion using values of Table 1) evaluated on the grid of synthetic systems presented in Sect. 2."

The validation supporting the headline uses informed phase-space dispersion hyper-parameters equal to the Table 1 values that generated the synthetic systems. The paper notes this option cannot be the default because it is system-specific. Thus the <10% errors measure the accuracy of an oracle dispersion input, not the default method's ability to infer ages from expansion data. For real systems these dispersions must be estimated, so the headline statistic is conditional on an input that is not available in application and is partly constructed by the validation setup.

full rationale

The paper is not circular in the sense of a self-citation chain or a derivation that is identical to its assumptions by algebra; the Bayesian hierarchical model, the NUTS sampling, and the comparison against frequentist estimators are genuine. However, the central validation claim is partially circular and conditional: the age prior is centered on the true age and the phase-space dispersion priors are set to the true generating values, and the paper explicitly states that in 25-source systems the prior dominates. The abstract's '<10% error / 100% credibility' statement is therefore a property of the oracle prior configuration, not of the expansion-rate likelihood alone. The paper is transparent about this in Sect. 4.2 and Fig. 11, where a shifted prior produces 10-15% errors and an HDI excluding the true age, and Sect. 6.2 warns that the prior is the most important source of bias. Because the headline is presented without that condition, the validation partially reduces to its own input; this is a partial circularity in the validation protocol rather than a fully self-referential derivation. Score 6 reflects one or more 'predictions' that are statistically forced by prior inputs.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The central quantitative claims rest on hand-chosen priors (age, dispersion, κσ) whose values in the validation are set to the truth, plus the standard expansion and Gaussian phase-space assumptions. The free parameters are therefore the main sensitivity lever, and the axioms are the usual dynamical-dating assumptions the authors themselves criticize in Appendix A.

free parameters (5)

Age prior hyperparameters (µτ, στ) = µτ=23 Myr, στ=8 Myr for β-Pictoris template
The prior center and width are taken from a literature age estimate and directly control the posterior; sensitivity tests show ±5 Myr shifts cause 10-15% errors.
Informed phase-space dispersion prior hyperparameters = σXYZ=(16.0,13.2,7.4) pc, σUVW=(1.5,0.5,0.7) km/s (β-Pic); (1,1,1) pc and km/s (Taurus)
In the main validation the dispersion priors are set to the true values used to generate the data (Table 1), which inflates performance.
κσ prior scale (Exponential) = 0.01 km/s/pc
Chosen by sensitivity analysis in Sect. 4.1 to minimize age errors; three values 0.001, 0.005, 0.01 were tested.
GGL age prior shape parameters (p, f*) = p=10 with scale factor f*; alternative p=1.19
Parameters of the Generalised Gamma Left prior tuned so that mode and variance match the literature age; the choice of skewness affects the posterior.
κν prior (Gamma α=2, β=1) = mode at dof=1
Fixed following Juárez & Steel (2010); affects StudentT tail weight for non-isotropic expansion cases.

assumptions (5)

domain assumption Young stellar associations are expanding at a constant rate since birth (Assumptions 1-3).
Appendix A lists these as assumptions; if a system is bound or has had a later event that changed expansion, the inferred age is wrong.
domain assumption Phase-space distribution is multivariate Gaussian or a Gaussian mixture with mutual Mahalanobis distance below two (Assumption 4).
This limits the method to single-population, roughly Gaussian systems and excludes complex substructure unless explicitly modeled.
domain assumption Gaia-like observational uncertainties are Gaussian and correctly modeled by PyGaia.
The simulation pipeline assumes univariate normal observational errors with PyGaia variances; real data may have correlations not captured here.
standard math The Lindegren linear velocity field model with rotation and shear terms is adequate.
Used as the generative model in Eq. 2; the paper notes in Sect. 6.3 that higher-order fields would improve flexibility, especially for star-forming regions.
domain assumption Unresolved binary astrometry corresponds to the center of mass and their radial velocities are unusable (Assumption 5).
The authors state this is 'currently the best we can do' and plan future filtering; binary contamination is a known source of bias.

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Cite this review

Pith. "Pith review of Going Bayesian on the ages of nearby young stellar systems I. The expansion rate method." pith.science (2026). https://pith.science/paper/RXLDSIXA

@misc{pith2026250605110,
  author       = {Pith},
  title        = {Pith review of: Going Bayesian on the ages of nearby young stellar systems I. The expansion rate method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RXLDSIXA}},
  note         = {Machine review of arXiv:2506.05110}
}

read the original abstract

Context. Determining the ages of young stellar systems is fundamental to test and validate current star-formation theories. Aims. We aim at developing a Bayesian version of the expansion rate method that incorporates the a priori knowledge on the stellar system's age and solves some of the caveats of the traditional frequentist approach. Methods. We upgrade an existing Bayesian hierarchical model with additional parameter hierarchies that include, amongst others, the system's age. For this later, we propose prior distributions inspired by literature works. Results. We validate our method on a set of extensive simulations mimicking the properties of real stellar systems. In stellar associations between 10 and 40 Myr and up to 150 pc the errors are <10%. In star forming regions up to 400 pc, the error can be as large as 80% at 3 Myr but it rapidly decreases with increasing age. Conclusions. The Bayesian expansion rate methodology that we present here offers several advantages over the traditional frequentist version. In particular, the Bayesian age estimator is more robust and credible than the commonly used the frequentist ones. This new Bayesian expansion rate method is made publicly available as a module of the free and open-source code Kalkayotl.

Figures

Figures reproduced from arXiv: 2506.05110 by the authors.

**Figure 1.** Comparison of our age prior distributions. The hyper-parameters (µτ and στ) were fixed to values that result in a distribution with mode and variance corresponding to those for the β-Pictoris stellar association, 23 ± 8 Myr (Lee et al. 2024). As mentioned above, the three expansion rate components κX, κY and κZ are drawn from the same statistical distribution, for which we use either a Normal or a StudentT, with lo… view at source ↗

**Figure 2.** Metrics for the age prior distributions for the generalised Gamma left (Default), truncated normal (TN), and generalised Gamma right (GGR). The lines and shaded regions depict the mean and standard deviation of the five synthetic simulations corresponding to each case. 40 60 Number of sources 5 0 5 10 15 20 % Metric = Error 40 60 Number of sources 10.0 12.5 15.0 17.5 20.0 22.5 25.0 % Metric = Uncertainty Experiment … view at source ↗

**Figure 3.** Metrics for the shifting and scaling experiments. For comparison purposes, the default configuration without shifting and scaling is also included. The lines and shaded regions depict the mean and standard deviation of the five synthetic simulations corresponding to each case. To save space, the credibility, which is always 100%, is not shown [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 5.** Figure 5: Metrics for the non-isotropic expansion experiments. The lines and shaded regions depict the mean and standard deviation of the five synthetic simulations corresponding to each case. To save space, the credibility, which is always 100%, is not shown. The expansion rate…

**Figure 6.** Figure 6: shows the metrics (as columns) of the Bayesian age estimator as a function of distance colour coded with the number of system’s sources and grouped in three ages (as rows). The line style shows the fraction of observed radial velocities. At first glance, the errors ar…

**Figure 7.** Figure 7: Age’s metrics of the synthetic star-forming regions. Captions as in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: Metrics of the classical frequentist age estimator applied to the datasets of synthetic associations. The lines and shaded regions depict the mean and standard deviation of the ten synthetic simulations corresponding to each case. of the frequentist estimators vary gre…

**Figure 9.** Figure 9: Metrics of the robust frequentist age estimator applied to the datasets of synthetic associations. Captions as in [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: Prior check plot of the age parameter (internally called 6D::age, in Myr) computed based of kernel density estimations from samples of the prior (µτ=23 Myr and στ=8 Myr) and posterior distributions for the β-Pictoris template with 25 sources and fully observed radial …

**Figure 11.** Figure 11: Examples of the prior’s influence on the posterior for the case of a realization of the β−Pictoris’ template with 25 sources and fully observed radial velocities. The panels show kernel density estimates of the prior (dotted lines) and posterior (solid lines) distribu…

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2019

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

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2015

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work page 2023

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2018

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R., & Patten , B

Barrado y Navascu \'e s , D., Stauffer , J. R., & Patten , B. M. 1999, , 522, L53

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