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arxiv: 2605.08544 · v1 · submitted 2026-05-08 · 🌌 astro-ph.EP · astro-ph.IM· astro-ph.SR

Recognition: 2 theorem links

· Lean Theorem

Bayesian Doppler Imaging: Simultaneous Inference of Surface Maps and Geometric Parameters

Yamato Ureshino , Hajime Kawahara , Hibiki Yama , Kento Masuda
Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:16 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IMastro-ph.SR
keywords Bayesian Doppler imagingsurface mappinggeometric parametersbrown dwarfLuhman 16BGaussian processspectral time seriesHamiltonian Monte Carlo
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The pith

A Bayesian pixel-based framework simultaneously infers surface brightness maps and geometric parameters such as inclination and equatorial rotation velocity from high-resolution spectral time series.

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

The paper develops a fully Bayesian Doppler imaging method that models the surface as a pixel grid with a Gaussian process prior on intensities. Geometric parameters including inclination and rotation speed are treated as nonlinear variables sampled by Hamiltonian Monte Carlo, while linear map coefficients are marginalized analytically. Validation on synthetic data shows the approach recovers longitudes of large-scale features and constrains the geometry parameters, and the method is then applied to VLT/CRIRES observations of the brown dwarf Luhman 16B. A reader would care because the joint inference supplies uncertainty estimates on both the map and the geometry without relying on fixed literature values for rotation or inclination.

Core claim

We present a fully Bayesian, pixel-based Doppler imaging framework that enables the simultaneous inference of surface brightness maps and geometric parameters, including the inclination i and equatorial rotation velocity v_rot, from high-resolution spectral time series. We treat the inference as a Bayesian linear inverse problem conditioned on nonlinear geometric parameters. The surface map is modeled as a Gaussian Process prior over pixel intensities, introducing a characteristic spatial scale that sets the map resolution. This allows analytical marginalization of the linear coefficients and efficient sampling of the nonlinear parameters with Hamiltonian Monte Carlo. Validation with synetht

What carries the argument

Gaussian Process prior over pixel intensities in a Bayesian linear inverse problem, which permits analytical marginalization of map coefficients and Hamiltonian Monte Carlo sampling of nonlinear geometric parameters.

Where Pith is reading between the lines

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

  • The joint-inference approach could be applied to other rapidly rotating objects to reduce biases that arise when geometry is fixed in advance.
  • The noted limited latitudinal sensitivity implies that Doppler imaging alone may always require supplementary data types to resolve features near the poles.
  • Public release of the code allows direct tests on new spectral datasets to check whether recovered dark regions persist under varied GP length scales.
  • If the derived radius from v_rot and i is combined with evolutionary models for other brown dwarfs, it could tighten constraints on their internal structure.

Load-bearing premise

The surface map is modeled as a Gaussian Process prior over pixel intensities that introduces a characteristic spatial scale setting the map resolution, and the method recovers features under the adopted model assumptions including the limited latitudinal sensitivity intrinsic to Doppler imaging.

What would settle it

Independent measurements of Luhman 16B's inclination or equatorial rotation velocity falling outside the reported ranges of 48.7 to 75.3 degrees and 28.1 to 36.5 km/s would challenge the joint inference.

Figures

Figures reproduced from arXiv: 2605.08544 by Hajime Kawahara, Hibiki Yama, Kento Masuda, Yamato Ureshino.

Figure 1
Figure 1. Figure 1: Geometry of the coordinate transformation when the rotational phase is φk = 0. The radial velocity of the j-th component at the k-th phase is given by (see Appendix A) vlos = vrot cos α sin θ ∗ j sin ϕ ∗ jk. (4) The corresponding Doppler factor affecting the wave￾length is Djk = 1 + β p 1 − β 2 , (5) where β is the line-of-sight velocity in units of the speed of light: β = vlos(θ ∗ j , ϕ∗ jk) c . (6) The s… view at source ↗
Figure 2
Figure 2. Figure 2: Joint posterior distributions of (i, vrot) inferred from synthetic spectra for each of the true inclinations i = 10◦ , 20◦ , . . . , 80◦ using Map 1. Blue dashed lines indicate the injected (true) values, red points show the posterior medians, and thin gray curves denote the locus vrot sin i = 10 km s−1 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Joint posterior distributions of (i, vrot) inferred from synthetic spectra at a fixed inclination of i = 40◦ for Maps 1, 2, and 3. Blue dashed lines indicate the injected (true) values, red points show the posterior medians, and thin gray curves denote the locus vrot sin i = 10 km s−1 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Posterior mean (top) and uncertainty (bottom) surface maps inferred from synthetic spectra. The rows show the results for Map 1, Map 2, and Map 3, respectively. The first column presents the ground truth, while the remaining columns display the reconstructions at inclinations of i = 10◦ , 40◦ , and 70◦ , shown on a common color scale within each row [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the joint posterior distribution of vrot and i, which exhibits the strongest correlation among the geometric parameters. Despite the conservative prior adopted for the real-data application, both pa￾rameters are well constrained by the time-resolved spec￾ [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reconstructed limb-darkening profiles derived from the posterior samples of the quadratic coefficients (q1, q2). The red line represents the median, while the thin gray lines show 1000 individual profiles drawn from the pos￾terior distribution, illustrating the uncertainty. For com￾parison, the blue dashed line shows the median profile ob￾tained from a separate preliminary run assuming a linear limb-darken… view at source ↗
Figure 7
Figure 7. Figure 7: Corner plot showing the one- and two-dimensional posterior distributions for all nonlinear parameters, excluding the regularization weight wk. The diagonal panels display the marginalized probability density for each parameter, while the off-diagonal panels show the joint distributions. The vertical lines in the diagonal panels indicate the 16th, 50th, and 84th percentiles [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 8
Figure 8. Figure 8: Posterior mean (top) and uncertainty (bottom) maps for Luhman 16B. Doppler imaging model for describing the spectral vari￾ability of Luhman 16B. 5. DISCUSSION 5.1. Surface Inhomogeneity and Comparison with Previous Work Our Bayesian Doppler imaging analysis has revealed a significant surface inhomogeneity on Luhman 16B. The posterior mean map ( [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Spectral fit and residuals for Luhman 16B. Top panel: Comparison between the observed spectral time series (black) and the model reconstruction based on the posterior median parameters (red). The spectra are vertically offset for clarity. Bottom panel: Two-dimensional map of the residuals (observation minus model) normalized by the inferred noise amplitude σ¯d. The color scale covers the range of ±3 ¯σd. T… view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of surface maps. (Top) Our pos￾terior mean surface map inferred from Luhman 16B data, shifted by 180◦ in longitude relative to [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A representative sample of the surface intensity map drawn from the posterior distribution. While the dark region observed in the posterior mean map is also visible here, the map exhibits high-frequency granular noise. This pixel-scale fluctuation is partly attributable to the jitter term included in the covariance matrix for numerical stability. The simultaneous inference of geometric parameters allows u… view at source ↗
Figure 12
Figure 12. Figure 12: Posterior probability distribution of the radius of Luhman 16B. The radius is derived from the joint poste￾rior samples of the equatorial rotation velocity vrot and the rotation period P using the relation R = vrotP/(2π). Regarding the limb darkening, our inferred quadratic coefficients (q1 = 0.81+0.13 −0.22, q2 = 0.59+0.16 −0.13) are broadly consistent with those derived from the atmospheric re￾trieval b… view at source ↗
Figure 13
Figure 13. Figure 13: Example of linear interpolation for Nwav = 6 and D = 3/4. The (l, l′ ) element of C (jk) , denoted c (jk) ll′ , is constructed as follows. For each wavelength grid point λl (l = 1, 2, . . . , Nwav) with spacing ∆λ, define ℓ (jk) l = λl/Djk − λ1 ∆λ + 1. (B11) The index is then clipped to the valid range 1 ≤ ˆℓ (jk) l ≤ Nwav by ˆℓ (jk) l = max 1, min(ℓ (jk) l , Nwav)  , (B12) and the interpolation coeffici… view at source ↗
read the original abstract

We present a fully Bayesian, pixel-based Doppler imaging framework that enables the simultaneous inference of surface brightness maps and geometric parameters, including the inclination $i$ and equatorial rotation velocity $v_{\mathrm{rot}}$, from high-resolution spectral time series. We treat the inference as a Bayesian linear inverse problem conditioned on nonlinear geometric parameters. The surface map is modeled as a Gaussian Process prior over pixel intensities, introducing a characteristic spatial scale that sets the map resolution. This allows analytical marginalization of the linear coefficients and efficient sampling of the nonlinear parameters with Hamiltonian Monte Carlo. {Validation with synthetic data demonstrates that our method recovers the longitudes of large-scale surface inhomogeneities and constrains $v_{\mathrm{rot}}$ and $i$ under the adopted model assumptions, while also revealing the limited latitudinal sensitivity intrinsic to Doppler imaging.} We applied this framework to high-resolution VLT/CRIRES observations of the brown dwarf Luhman 16B. Our analysis reveals a large-scale dark region at mid-latitudes, consistent with previous studies but now with spatially resolved uncertainty estimates. Furthermore, we successfully constrained the geometric parameters without fixing \(v_{\mathrm{rot}}\sin i\) or $i$ to literature values, deriving an inclination of $i = 61.0_{-12.3}^{+14.3}$ degrees and an equatorial rotation velocity of $v_{\mathrm{rot}} = 31.2_{-3.1}^{+5.3}~\mathrm{km\,s^{-1}}$. These results indicate a radius broadly consistent with evolutionary models and suggest a possible spin-axis misalignment under the assumption of comparable equatorial rotation velocities for the two components. Our code is publicly available under the MIT license.

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

2 major / 3 minor

Summary. The paper presents a fully Bayesian pixel-based Doppler imaging framework for jointly inferring surface brightness maps (via Gaussian Process priors over pixels) and nonlinear geometric parameters (inclination i and equatorial rotation velocity v_rot) from high-resolution spectral time series. Linear map coefficients are analytically marginalized for fixed geometry, with the nonlinear parameters sampled via Hamiltonian Monte Carlo. Synthetic data validation recovers longitudes of inhomogeneities and constrains geometry under the adopted model assumptions (including a single characteristic GP spatial scale and limited latitudinal sensitivity), while the application to VLT/CRIRES observations of Luhman 16B yields a mid-latitude dark region with spatially resolved uncertainties plus i = 61.0_{-12.3}^{+14.3} deg and v_rot = 31.2_{-3.1}^{+5.3} km s^{-1}. The code is made publicly available.

Significance. If the central results hold, the work provides a meaningful advance in Doppler imaging by enabling simultaneous posterior inference of maps and geometry without fixing v sin i or i a priori, along with spatially resolved uncertainty estimates. The synthetic validation and public code are strengths that support reproducibility and falsifiability. The Luhman 16B application demonstrates consistency with prior studies while adding quantitative uncertainties, with potential impact for atmospheric studies of brown dwarfs and directly imaged exoplanets.

major comments (2)
  1. [§4] §4 (Synthetic Validation): Recovery of v_rot and i is shown only for surface maps generated with the same fixed GP characteristic spatial scale used in the inference; no mismatch tests (e.g., injected maps with different correlation lengths or non-GP structure) are presented. This is load-bearing for the claim that geometry is robustly constrained independently of the prior scale choice, as the forward operator and marginalized likelihood could shift under realistic model mismatch.
  2. [§5.3] §5.3 (Luhman 16B results): The reported posterior for i (with ~13° uncertainties) is presented as a data-driven constraint, but the manuscript notes intrinsic latitudinal insensitivity of Doppler imaging. Without a prior-only comparison, information-gain metric, or explicit sensitivity test to the GP scale, it remains unclear whether the i posterior is meaningfully informed by the data or largely prior-dominated.
minor comments (3)
  1. [Abstract and §3] The abstract and §3 should more explicitly state whether the GP characteristic spatial scale is fixed a priori or optimized/marginalized, and how its value was chosen for the Luhman 16B analysis.
  2. [Figure 4] Figure 4 or equivalent (posterior maps): The uncertainty visualization would be clearer with an additional panel or colorbar showing the ratio of posterior standard deviation to prior standard deviation to highlight data-informed regions.
  3. [§2] Notation in the forward model (likely Eq. 5-7): The projection and line-of-sight velocity operators could be defined with explicit symbols for the pixel grid and Doppler shift to improve readability for readers outside the subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation for minor revision. We address each major comment below with proposed changes to the manuscript.

read point-by-point responses

  1. Referee: §4 (Synthetic Validation): Recovery of v_rot and i is shown only for surface maps generated with the same fixed GP characteristic spatial scale used in the inference; no mismatch tests (e.g., injected maps with different correlation lengths or non-GP structure) are presented. This is load-bearing for the claim that geometry is robustly constrained independently of the prior scale choice, as the forward operator and marginalized likelihood could shift under realistic model mismatch.

    Authors: We agree that mismatch tests would strengthen the validation. However, the manuscript already qualifies all synthetic results as holding 'under the adopted model assumptions,' which explicitly includes the single fixed GP spatial scale. We do not claim that geometry is constrained independently of the prior scale. Performing full mismatch tests with non-GP structures or varied correlation lengths would require substantial additional modeling and computation beyond the scope of this paper. In revision we will expand §4 with a discussion of how the marginalized likelihood depends on the GP hyperparameter and note that exploring model mismatch is an important direction for future work. revision: partial

  2. Referee: §5.3 (Luhman 16B results): The reported posterior for i (with ~13° uncertainties) is presented as a data-driven constraint, but the manuscript notes intrinsic latitudinal insensitivity of Doppler imaging. Without a prior-only comparison, information-gain metric, or explicit sensitivity test to the GP scale, it remains unclear whether the i posterior is meaningfully informed by the data or largely prior-dominated.

    Authors: The broad uncertainties on i (~13°) already signal the limited latitudinal sensitivity of Doppler imaging, as stated in the text. To clarify the data contribution, we will add to the revised §5.3 a direct overlay of the prior and posterior distributions for both i and v_rot, providing a visual information-gain assessment. We will also include a short sensitivity test showing how the geometric posteriors for Luhman 16B change when the GP characteristic scale is varied around the adopted value. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The framework models the surface map via a GP prior with a fixed characteristic spatial scale (a hyperparameter) to enable analytical marginalization of linear pixel coefficients for fixed nonlinear geometry (i, v_rot), followed by HMC sampling. Synthetic validation recovers injected features only under the stated model assumptions, which is the expected behavior for a correctly specified forward model rather than a tautology. Real-data posteriors on i and v_rot are presented as independent outputs from the marginalized likelihood applied to VLT/CRIRES spectra, without any reduction of the target quantities to quantities defined solely in terms of the fitted parameters or prior self-citations. No self-definitional, fitted-input-as-prediction, or load-bearing self-citation steps appear in the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Gaussian Process prior for pixel intensities and the standard Doppler imaging forward model; the main free parameter is the GP spatial scale, with no new physical entities postulated.

free parameters (1)
  • characteristic spatial scale
    Sets the map resolution in the Gaussian Process prior over pixel intensities
axioms (1)
  • domain assumption The surface map can be modeled as a Gaussian Process prior over pixel intensities
    This prior introduces the spatial scale and enables analytical marginalization of linear coefficients

pith-pipeline@v0.9.0 · 5620 in / 1358 out tokens · 75850 ms · 2026-05-12T01:16:17.962140+00:00 · methodology

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Works this paper leans on

47 extracted references · 47 canonical work pages · 1 internal anchor

  1. [1]

    B., Knutson, H

    Agol, E., Cowan, N. B., Knutson, H. A., et al. 2010, ApJ, 721, 1861, doi: 10.1088/0004-637X/721/2/1861

    work page doi:10.1088/0004-637x/721/2/1861 2010

  2. [2]

    2024, MNRAS, 532, 1361, doi: 10.1093/mnras/stae1549

    Aizawa, M., Muto, T., & Momose, M. 2024, MNRAS, 532, 1361, doi: 10.1093/mnras/stae1549

    work page doi:10.1093/mnras/stae1549 2024

  3. [3]

    Apai, D., Nardiello, D., & Bedin, L. R. 2021, ApJ, 906, 64, doi: 10.3847/1538-4357/abcb97 Artigau, ´E. 2018, in Handbook of Exoplanets, ed. H. J. Deeg & J. A. Belmonte (Springer International Publishing AG), 94, doi: 10.1007/978-3-319-55333-7 94 Asensio Ramos, A., D´ ıaz Baso, C. J., & Kochukhov, O. 2022, A&A, 658, A162, doi: 10.1051/0004-6361/202142027

    work page doi:10.3847/1538-4357/abcb97 2021

  4. [4]

    Hauschildt, P. H. 2003, A&A, 402, 701, doi: 10.1051/0004-6361:20030252

    work page doi:10.1051/0004-6361:20030252 2003

  5. [5]

    A Conceptual Introduction to Hamiltonian Monte Carlo

    Betancourt, M. 2017, arXiv e-prints, arXiv:1701.02434. https://arxiv.org/abs/1701.02434

    work page Pith review arXiv 2017

  6. [6]

    2017, The Astronomical Review, 13, 1, doi: 10.1080/21672857.2017.1303105

    Biller, B. 2017, The Astronomical Review, 13, 1, doi: 10.1080/21672857.2017.1303105

    work page doi:10.1080/21672857.2017.1303105 2017

  7. [7]

    2018, JAX: composable transformations of Python+NumPy programs, 0.2.5

    Bradbury, J., Frostig, R., Hawkins, P., et al. 2018, JAX: composable transformations of Python+NumPy programs, 0.2.5. http://github.com/google/jax

    work page 2018

  8. [8]

    C., Weiner Mansfield, M., Cubillos, P

    Challener, R. C., Weiner Mansfield, M., Cubillos, P. E., et al. 2025, Nature Astronomy, 9, 1821, doi: 10.1038/s41550-025-02666-9 22Ureshino et al

    work page doi:10.1038/s41550-025-02666-9 2025

  9. [9]

    A., Vos, J

    Chen, X., Biller, B. A., Vos, J. M., et al. 2024, Monthly Notices of the Royal Astronomical Society, 533, 3114–3143, doi: 10.1093/mnras/stae1995

    work page doi:10.1093/mnras/stae1995 2024

  10. [10]

    Collin, G. H. 2024, healjax: Port of healpix routines to JAX, 0.2.2. https://github.com/ghcollin/healjax

    work page 2024

  11. [11]

    B., & Agol, E

    Cowan, N. B., & Agol, E. 2008, ApJL, 678, L129, doi: 10.1086/588553

    work page doi:10.1086/588553 2008

  12. [12]

    Crossfield, I. J. M. 2014, A&A, 566, A130, doi: 10.1051/0004-6361/201423750

    work page doi:10.1051/0004-6361/201423750 2014

  13. [13]

    Crossfield, I. J. M., Biller, B., Schlieder, J. E., et al. 2014, Nature, 505, 654, doi: 10.1038/nature12955 de Wit, J., Gillon, M., Demory, B.-O., & Seager, S. 2012, A&A, 548, A128, doi: 10.1051/0004-6361/201219060

    work page doi:10.1038/nature12955 2014

  14. [14]

    Deutsch, A. J. 1958, in IAU Symposium, Vol. 6, Electromagnetic Phenomena in Cosmical Physics, ed. B. Lehnert, 209

    work page 1958

  15. [15]

    Duane, A

    Duane, S., Kennedy, A. D., Pendleton, B. J., & Roweth, D. 1987, Physics Letters B, 195, 216, doi: 10.1016/0370-2693(87)91197-X

    work page doi:10.1016/0370-2693(87)91197-x 1987

  16. [16]

    E., & Wehlau, W

    Falk, A. E., & Wehlau, W. H. 1974, ApJ, 192, 409, doi: 10.1086/153072

    work page doi:10.1086/153072 1974

  17. [17]

    2018, AJ, 156, 146, doi: 10.3847/1538-3881/aad775

    Robinson, T. 2018, AJ, 156, 146, doi: 10.3847/1538-3881/aad775

    work page doi:10.3847/1538-3881/aad775 2018

  18. [18]

    2016, The Journal of Open Source Software, 1, 24, doi: 10.21105/joss.00024

    Foreman-Mackey, D. 2016, Journal of Open Source Software, 1, 24, doi: 10.21105/joss.00024

    work page doi:10.21105/joss.00024 2016

  19. [19]

    2012, ApJ, 755, 101, doi: 10.1088/0004-637X/755/2/101

    Fujii, Y., & Kawahara, H. 2012, ApJ, 755, 101, doi: 10.1088/0004-637X/755/2/101

    work page doi:10.1088/0004-637x/755/2/101 2012

  20. [20]

    Gillon, M., Triaud, A. H. M. J., Jehin, E., et al. 2013, A&A, 555, L5, doi: 10.1051/0004-6361/201321620

    work page doi:10.1051/0004-6361/201321620 2013

  21. [21]

    Yagola, A. G. 1982, Soviet Ast., 26, 690

    work page 1982

  22. [22]

    Yagola, A. G. 1977, Soviet Astronomy Letters, 3, 147 G´ orski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759, doi: 10.1086/427976

    work page internal anchor Pith review doi:10.1086/427976 1977

  23. [23]

    J., Challener, R

    Hammond, M., Bell, T. J., Challener, R. C., et al. 2024, AJ, 168, 4, doi: 10.3847/1538-3881/ad434d

    work page doi:10.3847/1538-3881/ad434d 2024

  24. [24]

    D., & Gelman, A

    Hoffman, M. D., & Gelman, A. 2011, arXiv e-prints, arXiv:1111.4246. https://arxiv.org/abs/1111.4246

    work page arXiv 2011

  25. [25]

    2010, ApJ, 720, 1333, doi: 10.1088/0004-637X/720/2/1333 —

    Kawahara, H., & Fujii, Y. 2010, ApJ, 720, 1333, doi: 10.1088/0004-637X/720/2/1333 —. 2011, ApJL, 739, L62, doi: 10.1088/2041-8205/739/2/L62

    work page doi:10.1088/0004-637x/720/2/1333 2010

  26. [26]

    2022, The Astrophysical Journal Supplement Series, 258, 31, doi: 10.3847/1538-4365/ac3b4d

    Kawahara, H., Kawashima, Y., Masuda, K., et al. 2022, The Astrophysical Journal Supplement Series, 258, 31, doi: 10.3847/1538-4365/ac3b4d

    work page doi:10.3847/1538-4365/ac3b4d 2022

  27. [27]

    2020, ApJ, 900, 48, doi: 10.3847/1538-4357/aba95e

    Kawahara, H., & Masuda, K. 2020, ApJ, 900, 48, doi: 10.3847/1538-4357/aba95e

    work page doi:10.3847/1538-4357/aba95e 2020

  28. [28]

    2025, The Astrophysical Journal, 985, 263, doi: 10.3847/1538-4357/adcba2

    Kawahara, H., Kawashima, Y., Tada, S., et al. 2025, The Astrophysical Journal, 985, 263, doi: 10.3847/1538-4357/adcba2

    work page doi:10.3847/1538-4357/adcba2 2025

  29. [29]

    2025, ApJ, 988, 53, doi: 10.3847/1538-4357/adddbd

    Kawashima, Y., Kawahara, H., Kasagi, Y., et al. 2025, ApJ, 988, 53, doi: 10.3847/1538-4357/adddbd

    work page doi:10.3847/1538-4357/adddbd 2025

  30. [30]

    Khokhlova, V. L. 1976, Astronomische Nachrichten, 297, 203, doi: 10.1002/asna.19762970408

    work page doi:10.1002/asna.19762970408 1976

  31. [31]

    Kipping, D. M. 2013, MNRAS, 435, 2152, doi: 10.1093/mnras/stt1435

    work page doi:10.1093/mnras/stt1435 2013

  32. [32]

    A., Charbonneau, D., Allen, L

    Knutson, H. A., Charbonneau, D., Allen, L. E., et al. 2007, Nature, 447, 183, doi: 10.1038/nature05782

    work page doi:10.1038/nature05782 2007

  33. [33]

    R., Parmentier, V., et al

    Kreidberg, L., Line, M. R., Parmentier, V., et al. 2018, AJ, 156, 17, doi: 10.3847/1538-3881/aac3df

    work page doi:10.3847/1538-3881/aac3df 2018

  34. [34]

    2021, arXiv e-prints, arXiv:2110.06271, doi: 10.48550/arXiv.2110.06271

    Luger, R., Bedell, M., Foreman-Mackey, D., et al. 2021, arXiv e-prints, arXiv:2110.06271, doi: 10.48550/arXiv.2110.06271

    work page doi:10.48550/arxiv.2110.06271 2021

  35. [35]

    Majeau, C., Agol, E., & Cowan, N. B. 2012, ApJL, 747, L20, doi: 10.1088/2041-8205/747/2/L20 Pall´ e, E., Biazzo, K., Bolmont, E., et al. 2025, Experimental Astronomy, 59, 29, doi: 10.1007/s10686-025-10000-4

    work page doi:10.1088/2041-8205/747/2/l20 2012

  36. [37]

    Composable effects for flexible and acceler- ated probabilistic programming in numpyro.arXiv preprint arXiv:1912.11554, 2019

    Phan, D., Pradhan, N., & Jankowiak, M. 2019, arXiv preprint arXiv:1912.11554

    work page arXiv 2019

  37. [38]

    E., Tuominen, I., & Vilhu, O

    Piskunov, N. E., Tuominen, I., & Vilhu, O. 1990, A&A, 230, 363

    work page 1990

  38. [39]

    K., & Wang, J

    Plummer, M. K., & Wang, J. 2022, The Astrophysical Journal, 933, 163, doi: 10.3847/1538-4357/ac75b9 —. 2023, The Astrophysical Journal, 951, 101, doi: 10.3847/1538-4357/accd5d

    work page doi:10.3847/1538-4357/ac75b9 2022

  39. [40]

    Snellen, I. A. G., Brandl, B. R., de Kok, R. J., et al. 2014, Nature, 509, 63, doi: 10.1038/nature13253

    work page doi:10.1038/nature13253 2014

  40. [41]

    S., et al

    Tan, X., Zhang, X., Marley, M. S., et al. 2025, Science Advances, 11, doi: 10.1126/sciadv.adv3324

    work page doi:10.1126/sciadv.adv3324 2025

  41. [42]

    2005, Inverse Problem Theory and Methods for Model Parameter Estimation (the Society for Industrial and Applied Mathematics)

    Tarantola, A. 2005, Inverse Problem Theory and Methods for Model Parameter Estimation (the Society for Industrial and Applied Mathematics)

    work page 2005

  42. [43]

    2026, BayesianDI, 1.0.0, Zenodo, doi: 10.5281/zenodo.20024778

    Ureshino, Y. 2026, BayesianDI, 1.0.0, Zenodo, doi: 10.5281/zenodo.20024778

    work page doi:10.5281/zenodo.20024778 2026

  43. [44]

    S., & Penrod, G

    Vogt, S. S., & Penrod, G. D. 1983, Publications of the Astronomical Society of the Pacific, 95, 565, doi: 10.1086/131208

    work page doi:10.1086/131208 1983

  44. [45]

    , keywords =

    Vogt, S. S., Penrod, G. D., & Hatzes, A. P. 1987, ApJ, 321, 496, doi: 10.1086/165647

    work page doi:10.1086/165647 1987

  45. [46]

    W., M´erand, A., Thompson, W., et al

    Xuan, J. W., M´ erand, A., Thompson, W., et al. 2024, Nature, 634, 1070, doi: 10.1038/s41586-024-08064-x

    work page doi:10.1038/s41586-024-08064-x 2024

  46. [47]

    2025, arXiv e-prints, arXiv:2511.23018, doi: 10.48550/arXiv.2511.23018 Bayesian Doppler Imaging23

    Yama, H., Masuda, K., Kawashima, Y., & Kawahara, H. 2025, arXiv e-prints, arXiv:2511.23018, doi: 10.48550/arXiv.2511.23018 Bayesian Doppler Imaging23

    work page doi:10.48550/arxiv.2511.23018 2025

  47. [48]

    2019, The Journal of Open Source Software, 4, 1298, 10.21105/joss.01298

    Zonca, A., Singer, L., Lenz, D., et al. 2019, Journal of Open Source Software, 4, 1298, doi: 10.21105/joss.01298

    work page doi:10.21105/joss.01298 2019