Kalkayotl 2.0 Bayesian phase-space modelling of star-forming regions, stellar associations, and open clusters

A. Berihuete; H. Bouy; J. Olivares; Trevor Z. Dorn-Wallenstein

REVIEW 3 major objections 6 minor 55 references

Kalkayotl 2.0 Bayesian phase-space modelling of star-forming regions, stellar associations, and open clusters

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A single Bayesian hierarchical model can infer the 3D positions, velocities, expansion, and rotation of stellar systems with up to about 1000 stars from Gaia data, as demonstrated on beta Pictoris, the Hyades, and Praesepe.

desk verdict A solid, genuinely useful methods paper with real code and honest validation, but the rotation/expansion detection rules are never calibrated at null signal and the Hyades 2σ claim sits in the weak-signal regime the authors themselves caution about. read the letter →

arxiv 2411.16012 v1 pith:B2TVRPOQ submitted 2024-11-24 astro-ph.GA astro-ph.IMastro-ph.SR

J. Olivares , H. Bouy , Trevor Z. Dorn-Wallenstein , A. Berihuete This is my paper · ORCID

classification astro-ph.GAastro-ph.IMastro-ph.SR

keywords BayesianhierarchicalmodelGaiaastrometryphase-spaceinferencelinearvelocityfieldopenclustersstellarassociationsexpansionagerotation

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

This paper claims that the 3D structure and internal motions of star-forming regions, stellar associations, and open clusters can be inferred jointly from Gaia astrometry and radial velocities with one flexible Bayesian hierarchical model. The model, the multidimensional successor of the 1D Kalkayotl, treats each star's position and velocity as latent variables and lets the whole system be described by Gaussian, Student-t, or Gaussian-mixture populations, optionally with a linear velocity field that separates expansion, contraction, rotation, and velocity dispersion. Validated on synthetic clusters that mimic Gaia DR3, the method recovers population-level and per-star parameters with small errors and high credibility out to about 1.5 kpc. Applied to real benchmark systems, it reproduces literature values and yields an expansion age of $19.1\pm 1.0$ Myr for $\beta$ Pictoris, a $2\sigma$ rotation signal of $32\pm 11\ \mathrm{m\,s^{-1}\,pc^{-1}}$ in the Hyades, and the conclusion that Praesepe's reported rotation comes from its periphery. If the method is right, these kinematic diagnostics can be compared directly with star-formation theory predictions.

What carries the argument

The load-bearing object is a Bayesian hierarchical model with a linear velocity field. The system is described by a population-level location and covariance in 3D or 6D, while each star's position and velocity are latent variables; the velocity-field model sets $v_i = T(x_i - x_0) + D(v_0, \Sigma_v)$, where $T$ is a $3\times 3$ gradient tensor whose trace gives the expansion or contraction rate $|\kappa| = \tfrac{1}{3}\mathrm{tr}(T)$ and whose antisymmetric part gives the rotation vector $\omega$. The likelihood is a single high-dimensional multivariate Gaussian that keeps inter-source spatial correlations among Gaia parallaxes and proper motions, rather than assuming independent observations, and it naturally accommodates missing radial velocities. The prior structure uses flexible distributions on locations, scales, correlations, mixture weights, and degrees of freedom, together with a generalised-Gamma distance prior. The argument succeeds because this architecture lets expansion, rotation, and dispersion be separated from perspective effects and tested against zero with explicit posterior-interval criteria.

What would settle it

Run the same Hyades and beta Pictoris analyses with the default sky-uncertainty inflation and with a much smaller inflation factor, using longer chains so both runs converge; if the inferred rotation component or the expansion age shifts by more than the quoted uncertainties, the inflation assumption is refuted.

Watch

Extended reading notes

Core claim

The paper's central discovery is that a single hierarchical forward model can replace the usual two-step translation of Gaia measurements into physical space. In this model, individual stars' 3D positions and 3D velocities are source-level parameters drawn from a population-level distribution, and the likelihood is evaluated in the observed space by transforming proposed phase-space coordinates into astrometry and radial velocities. The new ingredient relative to earlier versions is a general linear velocity field: each star's velocity is the system's bulk velocity plus a 3x3 gradient tensor acting on the offset from the system centre, so expansion or contraction and rotation appear as distinct tensor components with objective posterior-based detectability criteria. The authors show, on realistic Gaia DR3-like simulations, that population parameters are recovered with relative errors typically below 10 to 20 percent and median credibility near 100 percent out to 1.5 kpc for the Gaussian, Student-t, and linear-field families, and out to 800 pc for Gaussian mixtures. On real data the model reproduces literature phase-space parameters for beta Pictoris, the Hyades, and Praesepe, and produces the new kinematic results listed in the abstract.

Load-bearing premise

The method depends on assuming that inflating the tiny Gaia sky-position uncertainties by a factor of 100,000 to 1,000,000 leaves the inferred parameters unchanged; if that heuristic is wrong, the reported positions, gradients, expansion ages, and rotation detections would all be biased.

Editorial extensions

If this is right

Kinematic diagnostics that are currently spread across different methods, namely expansion rate, contraction, rotation vector, velocity dispersion, and bulk motion, become outputs of one reproducible inference with stated posterior-interval detection criteria.
The method handles partial radial-velocity coverage, so the full astrometric sample is used instead of cropping to stars with radial velocities, avoiding bias toward bright members.
Users can fix any population parameter to a literature value and infer only the rest, making the code suitable for membership classification or single-star updates without rerunning the full system.
The validation limits give a practical applicability envelope: Gaussian, Student-t, and linear-field families work out to about 1.5 kpc and Gaussian mixtures out to about 800 pc for systems with up to about 1000 stars.
For real systems, the model yields outputs directly comparable to star-formation theory, such as the beta Pictoris expansion age of $19.1\pm 1.0$ Myr and the Hyades rotation of $32\pm 11\ \mathrm{m\,s^{-1}\,pc^{-1}}$.

Reading between the lines

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

Editorial inference: the heuristic sky-uncertainty inflation implies that individual stellar positions near the cluster centre are effectively down-weighted relative to the population-level prior, so the recovered source-level coordinates should be read as shrinkage estimates rather than as independent measurements.
Editorial inference: the 2-sigma Hyades rotation appears in different components in different reference frames, namely $\omega_z$ in ICRS and $\omega_x$ in Galactic coordinates, so a rotation with a fixed spatial orientation should have been consistent across frames; this suggests the signal is marginal until confirmed with higher-precision radial velocities.
Editorial inference: the same linear velocity tensor could be used to quantify differential rotation or shear in young associations and to place empirical upper limits on dissolution rates of sparse groups, at the cost of assuming the velocity field is truly linear over the system's size.
Editorial inference: a direct way to stress-test the method beyond the paper's simulations is to feed it a synthetic cluster with a known rotation of $30\ \mathrm{m\,s^{-1}\,pc^{-1}}$ at 50 pc and check that the recovered posterior interval contains the input; the paper's own Figure 3 suggests such a detection would be only marginal, so real claims at that level should be treated cautiously.

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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

Here's what you should know. This is a genuinely useful methods paper: it upgrades Kalkayotl from 1D distance inference to full 3D and 6D phase-space modelling with a linear velocity field, missing radial velocity handling, and corrections for Gaia angular correlations. The code is public, GPU-friendly, and the validation is honest about where it works. For common parameters (location, dispersions) it recovers true values with ~100% credibility up to 1.5 kpc (800 pc for GMM), and the real-data benchmarks on β-Pic, Hyades, and Praesepe reproduce literature values at the 2σ level. That part is solid.

The combination it offers—joint position-velocity inference with flexible distributions, missing data, and spatial correlations—is not in Lindegren et al. (2000) or Oh & Evans (2020). The extension is incremental but real, and shipping code and data is credit-earning.

The soft spot is real, and the stress-test note gets it right. Section 3 validates the linear velocity tensor only for C = 10, 50, 100 m s−1 pc−1. There is no null-signal (C=0) simulation, so the HDI-based detection rules in §2.3.2 (exclude zero from the HDI of |κ| or ω) have no measured false-positive rate. The paper itself warns that tensor entries ≲50 m s−1 pc−1 are unreliable, with relative errors ~50% for C=50 and ~200% for C=10, yet the headline Hyades rotation is 32±11 m s−1 pc−1, right in that regime. The 100% credibility values for tensor entries are driven by wide HDIs, not accuracy of point estimates.

The sky-uncertainty inflation heuristic (§2.6) is a second concern. Increasing sky errors by 10^5–10^6, and 10^7 for some β-Pic runs, is applied in every real-data run and never varied in validation. It may be benign, but it is a load-bearing choice for the exact weak-signal regime where the rotation claims live.

The Praesepe conclusion (rotation from the periphery) depends on post-hoc decontamination and tidal-radius cuts. The paper is transparent about these choices, and the cleaning does remove the signal, so the narrative is coherent, but a sensitivity analysis would strengthen it.

Who is this for? Anyone doing Bayesian phase-space inference of nearby clusters and associations with Gaia. It is a practical, well-documented tool with validation above the norm for this literature. The Hyades and Praesepe science claims are suggestive, not settled. This deserves a serious referee. I would send it out, with a request for a C=0 calibration and a sensitivity test on the sky-uncertainty scaling before publication.

Referee Report

3 major / 6 minor

Summary. The paper presents Kalkayotl 2.0, a Bayesian hierarchical modeling code for inferring 3D positions and 6D phase-space parameters of low-number stellar systems from Gaia astrometry and radial velocities. The model includes Gaussian, Student-T, and Gaussian mixture distributions, plus a linear velocity field that yields expansion, contraction, rotation, and shear. The code is validated on synthetic Gaia DR3-like datasets and applied to beta Pictoris, the Hyades, and Praesepe, recovering literature parameters and reporting an expansion age for beta Pic of 19.1 +/- 1.0 Myr, a 2-sigma rotation signal in the Hyades, and evidence that Praesepe's rotation arises from its periphery.

Significance. If the detection criteria are properly calibrated, this is a valuable open-source contribution: it handles missing radial velocities, models angular correlations in Gaia astrometry, and provides posterior predictive checks and decontamination tools. The synthetic validation is extensive and transparent about the poor recovery of the linear velocity tensor at low signal amplitudes. The real-data benchmarks reproduce literature values in most cases. However, the paper's headline detections rely on weak signals whose false-positive rate is not established, and the sky-uncertainty inflation heuristic used in all real-data runs is never subjected to a sensitivity analysis.

major comments (3)

[Sect. 3, Eq. (4)] The validation of the linear velocity field uses only tensors T = C.M with C in {10, 50, 100}; no null-signal (C = 0) simulation is reported. The detection criteria in Sect. 2.3.2, which declare expansion or contraction if the HDI of |kappa| excludes zero and rotation if the HDI of any omega component excludes zero, are therefore never tested for false-positive rate. This is load-bearing because the headline results, particularly the 2-sigma Hyades rotation (Table 4) and the Praesepe periphery conclusion (Sect. 4.3), are weak signals in the regime where Sect. 3.1.2 reports relative errors of about 50% for C=50 and about 200% for C=10. I request a null-signal validation: simulate clusters with T=0, run the same pipeline, and report the fraction of runs in which the HDI criteria falsely declare expansion or rotation, as a function of distance and number of stars.
[Sect. 2.6] The heuristic of inflating Gaia sky-position uncertainties by factors of 10^5 to 10^6 (and 10^7 for the beta Pic linear runs) is applied to every real-data run, but its effect on the posterior is never tested. The claim that this has negligible impact on the recovered parameter values is not supported by any sensitivity analysis. Because the likelihood is altered for all astrometric features, this could bias the inferred positions, velocity gradients, expansion ages, and rotation detections. I request a sensitivity test on synthetic data in which the scaling factor is varied (for example, 1, 10^2, 10^4, 10^6, and 10^7) and the recovery of the linear tensor and population parameters is compared across these settings.
[Sect. 3.1.2] The credibility metric, defined as the fraction of simulations in which the true value falls in the 95% HDI, reaches 100% for the tensor entries for all C values, including C=10 where the relative error is up to 200%. This shows that the credibility metric is insensitive in the weak-signal regime because the HDIs are extremely wide. The paper should report the bias and HDI width explicitly for the tensor entries, and it should use null-signal simulations to quantify the false-positive rate of the detection criteria. Without this, the 100% credibility values are potentially misleading as evidence of good performance in the regime relevant to the paper's main detections.

minor comments (6)

[Table 1 and Sect. 2.5] The default hyper-parameter for the Student-T degrees of freedom is given as nu_beta = 10 in Table 1 but as nu_beta = 0.1 in the text; please reconcile these values.
[Tables 4 and 7] The parameters w1 through w5 are reported in these tables but are never defined in the text or in Eq. (3); if they are the five components of the traceless symmetric shear tensor, please state this explicitly and give the mapping to the tensor entries.
[Sect. 4.1 and Sect. 5] Section 4.1 reports expansion detections at 12 sigma, 3 sigma, and 1 sigma in the X, Y, and Z directions, while the conclusion states that the model detects expansion at the 2 sigma level; please clarify which quantity, such as the combined |kappa|, the 2 sigma statement refers to.
[Sect. 1] The sentence 'we exemplify the use of Kalakyotl by applying it to beta-Pictoris' contains a typo in the code name; it should read 'Kalkayotl'.
[Sect. 2.3.2] The diagonal entries of T are labeled kappa_x, kappa_y, kappa_z in Eq. (3) but are subsequently referred to as kappa_0, kappa_1, and kappa_2; please unify the notation.
[Abstract and Sect. 3] The abstract states that the code is intended for systems with up to about 1000 stars, but the synthetic validation grid only goes up to N_stars = 400; please either extend the validation or add a statement about expected performance up to 1000 stars based on the real-data applications.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: simulation validation and age inversion are transparent, with external real-data benchmarks anchoring the central claims.

full rationale

No significant circularity. The hierarchical forward model is not constructed from the results it reports: the tensor entries T, the expansion parameter kappa, and the rotation vector omega are free population parameters with zero-centred priors (Sect. 2.5), and the likelihood is built from the Gaia observables (Sect. 2.6). The Sect. 3 simulations are self-consistency checks: Amasijo draws synthetic clusters from specified true parameters and the code recovers them, so they validate the implementation rather than establishing an input-output equivalence. The beta Pictoris expansion age is an explicit inversion, tau = gamma^-1 kappa^-1, of the fitted expansion rate (Sect. 4.1); the paper itself calls it a 'simple inversion' and benchmarks it against independent trace-forward and trace-back ages, so it is a transparent posterior transformation rather than a disguised input. The Hyades 2-sigma rotation signal and the Praesepe periphery conclusion are outputs of the HDI detection criteria and of the cleaned-versus-uncleaned sample comparison, respectively, not identities imposed by the model definitions. Self-citations to Paper I and to Olivares et al. 2023a/b are normal method and application extensions and are not the load-bearing justification of the astrophysical claims. The sky-uncertainty inflation heuristic (Sect. 2.6) and the absence of a C=0 null-signal simulation are validation gaps and correctness risks, but they are not circularity under the defined patterns.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The model depends on standard statistical distribution choices (Gaussian, Student-T, GMM), a linear velocity gradient model inherited from Lindegren et al. (2000), and several prior hyperparameters. The most consequential hand-set inputs are the sky uncertainty scaling factor (10^6 default), the decontamination field scale (20 pc, 5 km/s), and the default prior widths for the velocity gradient entries. No new entities are introduced.

free parameters (3)

sky_uncertainty_scaling_factor = 10^6 (default), 10^7 used for beta Pic linear runs
Heuristic multiplier applied to Gaia sky-position uncertainties to ease NUTS convergence; asserted to have negligible impact, but no sensitivity analysis is shown. It affects all real-data results.
field_scale (decontamination FGMM) = 20 pc (position), 5 km/s (velocity)
User-provided standard deviations for the field component in the FGMM decontamination model; the Praesepe conclusion about rotation arising from the periphery depends on this choice and the subsequent tidal-radius cut.
prior scale for kappa and Omega (sigma_kappa, sigma_Omega) = 0.1 km/s/pc (default)
Zero-centred Gaussian prior widths for the velocity gradient tensor entries; they influence detectability of low signals and the posterior for the Hyades rotation.

assumptions (4)

domain assumption Gaia measurements are normally distributed (Assumption 1)
The likelihood is a multivariate Gaussian; deviations from normality (e.g., outliers, systematic errors) are handled only via the Student-T option in the population model, not in the measurement model.
domain assumption Parallax and proper-motion spatial correlations follow Eqs. 24-25 of Lindegren et al. 2021 (Assumption 3)
The inter-source covariance is populated using these published empirical correlation functions; if the Gaia DR3 correlations are inaccurate or misapplied, the uncertainties and posterior widths would be wrong.
domain assumption The input list of members is unbiased (Assumption 4, as in Paper I)
For the 3D/6D models the paper claims this assumption is no longer needed, yet the real-data applications rely on externally defined member lists (Couture, Miret-Roig, Crundall, Hao, GG, Jadhav), which may carry selection biases that affect the recovered system parameters.
ad hoc to paper The linear velocity field v_i = T dot (x_i - x_0) + D(v_0, Sigma_v) is an adequate model for internal kinematics (Eq. 2)
This is the core kinematic model; real clusters with tidal tails, substructure, or differential rotation may not follow a single linear gradient, so the inferred kappa and omega are average gradients over the modelled region.

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Cite this review

Pith. "Pith review of Kalkayotl 2.0 Bayesian phase-space modelling of star-forming regions, stellar associations, and open clusters." pith.science (2026). https://pith.science/paper/B2TVRPOQ

@misc{pith2026241116012,
  author       = {Pith},
  title        = {Pith review of: Kalkayotl 2.0 Bayesian phase-space modelling of star-forming regions, stellar associations, and open clusters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2TVRPOQ}},
  note         = {Machine review of arXiv:2411.16012}
}

abstract

Context: Star-forming regions, stellar associations, and open clusters are fundamental stellar systems where predictions from star-formation theories can be robustly contrasted with observations. Aims: We aim to provide the astrophysical community with a free and open-source code to infer the phase-space (i.e. positions and velocities) parameters of stellar systems with $\lesssim$1000 stars based on \textit{Gaia} astrometry and possibly observed radial velocities. Methods: We upgrade an existing Bayesian hierarchical model and extend it to model 3D (positions) and 6D (positions and velocities) stellar coordinates and system parameters with a flexible variety of statistical models, including a linear velocity field. This velocity field allows for the inference of internal kinematics, including expansion, contraction, and rotation. Results: We extensively validated our statistical models using realistic simulations that mimic the properties of the \textit{Gaia} Data Release 3. We applied \textit{Kalkayotl} to $\beta$-Pictoris, the Hyades, and Praesepe, recovering parameter values compatible with those from the literature. In particular, we found an expansion age of $19.1\pm1.0$ Myr for $\beta$-Pictoris and rotational signal of $32\!\pm\!11\,\rm{m\,s^{-1}\,pc^{-1}}$ for the Hyades and that Praesepe's rotation reported in the literature comes from its periphery. Conclusions: The robust and flexible Bayesian hierarchical model that we make publicly available here represents a step forward in the statistical modelling of stellar systems. The products it delivers, such as expansion, contraction, rotation, and velocity dispersions, can be directly contrasted with predictions from star-formation theories.

Figures

Figures reproduced from arXiv: 2411.16012 by the authors.

**Figure 1.** Relative error of the population-level parameters common to all our models as a function of distance. The lines and shaded regions depict the mean and standard deviation of the synthetic clusters with 100 stars. negligible credibility. From these figures, we draw the following conclusions. First, the errors and uncertainties are isotropic showing similar values in the position and in the velocity coordinates. The A… view at source ↗

**Figure 2.** Relative uncertainty of the population-level parameters common to all our models as a function of distance. Captions as in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Signal-to-noise ratio of the entries in the linear velocity tensor as a function of the distance and the number of stars for the three values of the C constant in units of metre per second per parsec (see Eq. 4). The lines and shaded regions depict the median and standard deviation of the synthetic simulations with varying random seed. Usually, the detection of a signal is based on its significance level, with 3σ to… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Relative error of the source-level parameters as a function of distance. Captions as in [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Relative uncertainty of the source-level parameters. Captions as in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: One-to-one comparison of the source-level parameters inferred here and those reported by Couture et al. (2023) and Miret-Roig et al. (2020). The grey line depicts the identity relation. The root-mean-squared value of the differences is also shown on the lower right sid…

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Pith tools

Reviewed August 12, 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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2013, , 554, A101

Bouy , H., Bertin , E., Moraux , E., et al. 2013, , 554, A101

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