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This paper argues that how cross-band dependence is represented in a multi-output Gaussian process — directly through a matrix-valued covariance function or indirectly through latent processes and linear operators — is the primary modeling

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2026-08-01 07:27 UTC pith:R3SZ4Z55

load-bearing objection Useful, well-organized synthesis of two multi-output GP constructions, but Eq. (10) has a genuine phase error in the cross-spectrum that must be fixed before the paper becomes a citable general reference. the 3 major comments →

arxiv 2607.21431 v1 pith:R3SZ4Z55 submitted 2026-07-23 astro-ph.IM stat.AP

Modeling Dependence Structures in Astronomical Multi-Band Time Series Data via Multi-Output Gaussian Processes

Samata Das , Lishan Shi , Yasaman Hamayouni , Hyungsuk Tak , Jong-Hak Woo This is my paper
classification astro-ph.IM stat.AP MSC 62M1060G15
keywords multi-output Gaussian processesdamped random walkpower spectral densitycoherencereverberation mappingactive galactic nucleiseparable covariancelatent process
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper establishes that, in multi-output Gaussian process modeling of astronomical light curves, the representation of dependence among photometric bands is the central modeling decision, not just which kernel is used. It compares covariance-based specifications, which describe the stochastic properties of the observed bands directly, with latent-process specifications, which build the observed light curves from shared latent Gaussian processes via linear operators. The paper derives closed-form power spectral density matrices for both formulations, shows that under a separable damped random walk model the coherence between bands is frequency-independent and equals the squared correlation, and demonstrates on SDSS Stripe 82 and RM840 data that scientifically different conclusions can follow from nearly identical fits. A sympathetic reader would care because it clarifies when to report cross-spectra versus transfer functions, and what either choice implies about variability that is not directly observed.

Core claim

The central claim is that the choice between specifying a matrix-valued covariance function directly and inducing it through latent processes is a statistical modeling decision with scientific consequences, not a purely computational preference. Using multi-output damped random walk (DRW) models as the running example, the paper derives the power spectral density matrix for the separable covariance-based model, S_ij(ω) = ρ_ij σ_i σ_j τ² / (1 + ω²τ²), and for the latent-process model, S_ij(ω) = Ψ̂_i(ω) Ψ̂_j(ω) S_Z(ω). It shows that coherence simplifies to γ²_ij = ρ²_ij under separability, turning a frequency-dependent diagnostic into a single number per band pair, and it shows that for the SD

What carries the argument

The separable covariance construction K(t,t′) = D_σ R D_σ ⊗ k(t,t′), where R is a correlation matrix, D_σ holds band amplitudes, and k(t,t′) is a shared damped random walk kernel with a single timescale τ; together with the latent-process identity that convolution with a transfer function multiplies the latent power spectral density by the squared modulus of the transfer function's Fourier transform. The former licenses a valid multi-output Gaussian process and yields the coherence identity; the latter separates the physics (the operator) from the stochastic driver (the latent process) and shows that the mean lag appears only in the cross-spectrum phase, not in the marginal power spectral de

Load-bearing premise

The separable covariance model needs every photometric band to share the same damped random walk timescale τ, a premise chosen to make the covariance construction valid; if AGN variability genuinely has band-dependent timescales, the strong cross-band coherences the model reports are partly an artifact of that assumption.

What would settle it

Estimate the frequency-resolved coherence between the g and z bands of a Stripe 82 quasar using a Fourier-based periodogram; the separable model predicts it is flat across frequency at the fitted value ρ²_gz ≈ 0.71. A coherence that rises or falls significantly with frequency would refute the common-timescale assumption. A complementary simulation: generate light curves from a multivariate DRW with distinct τ per band, fit the separable model, and check whether the estimate of ρ² is biased upward.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Under the separable multi-output DRW model, coherence is constant across frequency and equals ρ²_ij, so one number per band pair summarizes cross-band linear dependence, and comparisons between band pairs become direct.
  • In continuum reverberation mapping, the transfer-function shape enters the emission-line power spectral density only through |Ψ̂(ω)|², so the mean lag is identified from the cross-spectrum phase while the width and shape are identified from the amplitude; the ΔAIC ≈ 11 favoring the Gaussian transfer function indicates a smoother delay distribution describes RM840 better.
  • Because the two transfer functions with equal delay variance fit nearly equally, the inferred latent DRW parameters are insensitive to the operator, and the differences appear at frequencies poorly constrained by the data, implying that denser-cadence surveys will sharpen the distinction.
  • The framework extends beyond DRW to Matérn, quasi-periodic, and CARMA processes for both formulations, so the dependence structure remains the primary modeling choice regardless of the chosen temporal kernel.

Where Pith is reading between the lines

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

  • A natural testable extension: estimate the frequency-resolved coherence for the five Stripe 82 bands using Fourier-based periodogram methods; the separable model predicts it is flat across frequency at each fitted ρ²_ij. If coherence varies with frequency, the common-timescale assumption is violated and state-space multivariate DRW models with band-dependent timescales should be considered.
  • The delay-variance-matched comparison (w_TH = √12 σ_G) isolates the effect of transfer-function shape while holding width equal, so the ΔAIC ≈ 11 reflects shape rather than overall width; repeating the comparison at several fixed widths would map where the statistical preference breaks down.
  • The paper's point that dependence structures govern unresolved variability implies that sparse-cadence surveys will be more sensitive to the assumed structure; a simulation study across survey cadences could turn this into an operational rule for choosing between the two formulations.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper develops a unified framework for multi-output Gaussian process modeling of astronomical multi-band time series, distinguishing covariance-based and latent-process formulations. It derives PSD matrices for separable multi-output DRW models (Eqs. 8–9), the coherence identity γ²_ij=ρ²_ij, and convolution-based latent-process spectra (Eq. 10), and illustrates the two approaches on an SDSS Stripe 82 quasar and SDSS-RM target RM840. The central claim is that how cross-band dependence is represented is a primary modeling decision that affects scientific interpretation even when fits are similar.

Significance. The paper’s central conceptual message — that the choice of dependence representation, not just the marginal kernel, shapes statistical and scientific interpretation — is important and well illustrated. The closed-form expressions for the separable DRW PSD matrix, the coherence simplification, and the time-domain covariance constructions in §2 are useful and transparent. The appendices are largely checkable, and the public GitHub code makes the analysis reproducible. There is no circularity: the PSDs are derived from assumed covariance/linear-operator structure, not fitted and relabeled as predictions. However, the claimed general cross-spectrum in Eq. (10) contains a conjugation error, and the application-level claims in §4.2 rest on fixed width parameters whose uncertainty is not propagated. These issues are correctable within the scope of the paper.

major comments (3)
  1. [Appendix B, Eq. (10)] The off-diagonal cross-spectrum is missing a complex conjugation. With the paper’s Fourier convention \hat Ψ_i(ω)=∫Ψ_i(u)e^{-iωu}du, the change-of-variable step in Eq. (B1) gives an inner factor e^{iωa}e^{-iωb}S_Z(ω), so S_ij(ω)=[∫Ψ_i(a)e^{iωa}da][∫Ψ_j(b)e^{-iωb}db]S_Z(ω)=\hat Ψ_i(-ω)\hat Ψ_j(ω)S_Z(ω). For real Ψ_i this equals \hat Ψ_i^*(ω)\hat Ψ_j(ω)S_Z(ω), not \hat Ψ_i(ω)\hat Ψ_j(ω)S_Z(ω). The false identity '∫Ψ_i(a)e^{iωa}da=\hat Ψ_i(ω)' is stated in Appendix B. For two shifted transfer functions Eq. (10) predicts cross-spectral phase -(τ_1+τ_2)ω instead of -(τ_2-τ_1)ω. Diagonal entries and the RM840 application (one identity continuum, one line) are unaffected, but the advertised general result and the Hermitian-symmetry statement S_ji(ω)=S_ij(ω) following Eq. (10) must be corrected.
  2. [§4.2, Table 4] The comparison between top-hat and Gaussian transfer functions fixes σ_G=5 d and w_TH=√12 σ_G after preliminary width estimates ran into the lower boundary. The reported Hessian SEs for τ_0 and the ΔAIC≈11 preference treat these widths as known constants, so the quoted lag precision and the model-selection gap do not reflect uncertainty in the width choice. A sensitivity analysis over a range of fixed widths, or a profile-likelihood treatment, is needed before claiming that the Gaussian form is statistically preferred.
  3. [§2.1, Tables 2–3] The separable covariance construction assumes a common DRW timescale τ across all five bands; the paper itself notes this is adopted to obtain a valid separable construction. This assumption forces the common break frequency in Fig. 3 and the constant coherence γ²_ij=ρ²_ij in Table 3. The high coherence values (0.71–0.99) are therefore partly structural consequences of the model rather than empirical estimates of band dependence. Given that the multivariate DRW of Hu & Tak (2020), cited in §5.1, allows band-dependent timescales, the authors should either add a comparison with a model allowing different τ_j or soften the interpretation of the coherence estimates.
minor comments (3)
  1. [§4.2] The statement that the lag difference 'differing by only 2.17 days (approximately 1.6% of the inferred lag)' is misleading because the reported SEs are 0.11 and 0.19 days; the difference is many standard errors.
  2. [§3, Eq. (10)] The notation \hat Ψ_i in Eq. (10) is not defined until Appendix B; define the Fourier convention in the main text before using it.
  3. [§5.3] The paper appropriately notes that uncertainty in derived quantities is not propagated; a one-sentence acknowledgment in §4.2 would improve transparency.

Circularity Check

0 steps flagged

No significant circularity: the spectral, coherence, and PSD results are mathematical consequences of explicitly stated covariance/operator models, not re-labeled fits.

full rationale

The derivation chain is self-contained. Appendix A obtains S_ij(omega) = rho_ij sigma_i sigma_j tau^2 / (1 + omega^2 tau^2) by a standard Fourier integral from the stated separable covariance K_ij(u) = rho_ij sigma_i sigma_j (tau/2) exp(-|u|/tau); this is algebra, not an empirical prediction. The coherence identity gamma^2_ij = rho_ij^2 in Section 3 is an immediate consequence of that PSD matrix, and Table 3 simply tabulates fitted correlations; it is not a fit disguised as a prediction. Appendix B derives the latent-process PSD matrix directly from the convolution definition and the stated Fourier convention. A separate technical concern exists about the cross-spectrum in Eq. (10): the step 'integral Psi_i(a) exp(i omega a) da = Psi_hat_i(omega)' in Appendix B is the conjugate of the stated convention for non-symmetric transfer functions, which would affect phase/lags for two reprocessed bands. That is a correctness/phase issue, not a circularity: the RM840 application uses one identity continuum and one line, and the marginal PSDs depend only on |Psi_hat|^2. The RM840 comparison fixes transfer-function widths only after reporting that boundary estimates are weakly identified, and then compares fitted likelihoods; the AIC difference is a model comparison on the training data, not a prediction whose validity is built into the fitted parameters. Self-citations (Hu & Tak 2020; Tak et al. 2017) are used only as pointers to alternative SDE/state-space models and computational methods, not as imported uniqueness theorems or as evidence for the paper's central claim. The paper candidly flags limitations: the common-timescale assumption is adopted for valid separability (Section 2.1), width parameters are fixed after preliminary boundary estimates (Section 4.2), and uncertainty in derived quantities is deferred (Section 5.3). These are limitations, not circular reductions. I find no step in which an output is identical by construction to an input or in which a load-bearing claim is justified only by the authors' own prior work.

Axiom & Free-Parameter Ledger

7 free parameters · 8 axioms · 0 invented entities

No new physical entities or new latent constructs are introduced: the latent DRW is a standard latent-variable modeling device, not an invented entity with independent falsifiable handles. All model parameters in the two illustrations are fitted by maximum likelihood or chosen by hand; the framework itself adds no free parameters beyond those of the standard GP machinery it adopts. The central derivations rely on standard Fourier/GP results plus three domain assumptions (known error variances, a specific latent process, a specific transfer-function family), of which the common-timescale separability premise is the most consequential.

free parameters (7)
  • Common DRW timescale τ (covariance-based) = 553.03 d (SE 340.73 d)
    MLE in §4.1, Table 2; shared by all five bands by the separability construction; weakly identified (SE ≈ 62% of estimate).
  • Per-band diffusion coefficients σ_j = σ_u=0.0102, σ_g=0.0037, σ_r=0.0041, σ_i=0.0029, σ_z=0.0029
    MLE in §4.1, Table 2; set short-timescale variability amplitude per band.
  • Cross-band correlation matrix R = ρ ∈ [0.84, 0.99] (Table 2)
    MLE in §4.1; defines the dependence structure; coherence γ²=ρ² follows from it.
  • Band means μ_j = 18.67–21.22 mag
    MLE in §4.1; nuisance parameters.
  • Latent DRW parameters (RM) = σ=0.16, τ≈51 d, α_ℓ≈129, μ_c≈8.06, μ_ℓ≈535
    MLE in §4.2, Table 4; nearly identical across tophat/Gaussian models.
  • Mean reverberation lags τ0 = 136.51 d (top-hat), 138.68 d (Gaussian)
    MLE in §4.2; the scientific target of the RM illustration; SEs 0.11–0.19 d.
  • Fixed transfer-function widths = σ_G=5 d; w_TH=√12·5≈17.3 d
    Hand-chosen in §4.2 after preliminary width estimates hit the lower bound 0.05; w_TH chosen to match delay variance of the Gaussian.
axioms (8)
  • standard math GPs are closed under deterministic linear transformations (linear mixing, convolution, derivatives)
    Invoked in §2.2 to guarantee the latent-process multi-output covariance is automatically positive definite.
  • standard math DRW covariance is a valid stationary positive-definite kernel with Lorentzian PSD
    Used throughout; PSD computed by the standard Fourier integral in Appendix A.
  • standard math Kronecker product of positive-definite matrices is positive definite
    §2.1 Eq. (5): justifies the separable model K = D_σ R D_σ ⊗ k(t,t′).
  • domain assumption Measurement errors are Gaussian with known variances δ_ij and independent of the signal
    §2.3: 'the corresponding measurement uncertainty δ_ij is known'; 'independent both across observations and from the latent GP.'
  • ad hoc to paper All photometric bands share one common DRW timescale τ
    §2.1: 'the common-timescale assumption is adopted to obtain a valid separable covariance construction' — load-bearing for the covariance-based model.
  • domain assumption Latent processes are mutually independent
    §2.2: Cov(Z_ℓ,Z_m)=0 for ℓ≠m, giving the simplified Eq. (6).
  • domain assumption Latent continuum is a zero-mean DRW and the line is a scaled lagged convolution of it
    §4.2: Z(t)~DRW(0,σ,τ); X_ℓ = μ_ℓ + α_ℓ ∫ Ψ(u)Z(t−u)du — the reverberation assumption.
  • ad hoc to paper Transfer functions are normalized (integrate to 1) with fixed widths
    §4.2: normalization imposed; widths fixed to isolate transfer-function shape.

pith-pipeline@v1.3.0-alltime-deepseek · 20678 in / 23418 out tokens · 199077 ms · 2026-08-01T07:27:59.720844+00:00 · methodology

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read the original abstract

Modern astronomical time-domain surveys routinely collect multi-band light curves that provide complementary information about the physical processes governing source variability. Gaussian processes (GPs) provide a flexible probabilistic framework for modeling irregularly sampled and noisy time-series data. While considerable attention has been devoted to developing covariance kernels for individual time series, comparatively less attention has been paid to the statistical representation of dependence among multiple photometric bands. In this work, we present a unified statistical framework for modeling such dependence structures using multi-output GPs. Within this framework, we consider two complementary formulations. The covariance-based formulation specifies dependence directly through matrix-valued covariance functions and emphasizes the stochastic properties of the observed light curves, including covariance functions and power spectral densities. In contrast, the latent-process formulation represents the observed light curves as transformations of latent GPs and emphasizes the physical mechanisms generating the observed dependence. To illustrate these formulations, we develop covariance-based and latent-process multi-output damped random walk models and derive their corresponding spectral representations. We further demonstrate the practical implications of dependence-structure modeling through applications to multi-band active galactic nucleus variability and continuum reverberation mapping. Rather than advocating a universally preferred formulation, this work provides a principled basis for selecting dependence structures according to the scientific objectives and clarifies how this choice influences the statistical characterization and scientific interpretation of stochastic variability in astronomical sources.

Figures

Figures reproduced from arXiv: 2607.21431 by Hyungsuk Tak, Jong-Hak Woo, Lishan Shi, Samata Das, Yasaman Hamayouni.

Figure 1
Figure 1. Figure 1: provides a conceptual overview of these two approaches. At each time t, the GP values in the g and r bands are stacked to form a multivariate Gaus￾sian random vector with a corresponding stacked mean vector and covariance matrix. The diagonal blocks of the covariance matrix describe the temporal depen￾dence within individual photometric bands, whereas the off-diagonal blocks characterize the dependence acr… view at source ↗
Figure 2
Figure 2. Figure 2: Five-band SDSS Stripe 82 light curves of the quasar RM 1540, selected from the catalog of 9258 spec￾troscopically confirmed quasars compiled by C. L. MacLeod et al. (2012). The observations are irregularly sampled and contain different numbers of measurements across the five photometric bands. The reported 1-σ measurement uncer￾tainties are smaller than the plotting symbols and are there￾fore not visible. … view at source ↗
Figure 3
Figure 3. Figure 3: Estimated marginal (upper panel) and cross-band (lower panel) power spectral densities under the covariance-based separable multi-output DRW model fitted to the SDSS Stripe 82 quasar (dbID: 1540). The marginal PSDs share a common break frequency determined by the es￾timated characteristic timescale, while their amplitudes dif￾fer according to the estimated diffusion coefficients. Like￾wise, the cross-band … view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: compares the fitted transfer functions under the two operator assumptions. Although both transfer functions are normalized to integrate to one, they redis￾tribute continuum variability differently in time, leading directly to different frequency responses and hence dif￾ferent induced dependence structures for the observed emission-line variability [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Conditional reconstructions of the Hα emission-line light curve for RM840 under the fixed-width top-hat (upper panel) and Gaussian (lower panel) transfer-function models. Points and error bars denote the observed Hα measurements, solid curves show the conditional means, and shaded regions represent pointwise 95% conditional intervals. The two models produce similar reconstructions over the observed time ra… view at source ↗
Figure 7
Figure 7. Figure 7: Frequency-domain comparison of the fixed-width top-hat and Gaussian transfer-function models. The upper panel displays the squared magnitudes of the cor￾responding Fourier transforms. The lower panel shows the base-10 logarithm of the ratio of the fitted Hα emission– line PSDs. The horizontal dashed line denotes equal filtered power under the two models. Values above zero indicate that the top-hat model pr… view at source ↗

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Works this paper leans on

48 extracted references · 6 canonical work pages

  1. [1]

    2023, Annual Review of Astronomy and Astrophysics, 61, 329, doi: 10.1146/annurev-astro-052920-103508 ´Alvarez, M

    Aigrain, S., & Foreman-Mackey, D. 2023, Annual Review of Astronomy and Astrophysics, 61, 329, doi: 10.1146/annurev-astro-052920-103508 ´Alvarez, M. A., Rosasco, L., & Lawrence, N. D. 2012, Foundations and Trends in Machine Learning, 4, 195, doi: 10.1561/2200000036

    doi:10.1146/annurev-astro-052920-103508 2023

  2. [2]

    C., et al

    Bellm, E. C., et al. 2019, PASP, 131, 018002, doi: 10.1088/1538-3873/aaecbe

    doi:10.1088/1538-3873/aaecbe 2019

  3. [3]

    2019, AJ, 158, 257, doi: 10.3847/1538-3881/ab5f2e

    Boone, K. 2019, AJ, 158, 257, doi: 10.3847/1538-3881/ab5f2e

    work page doi:10.3847/1538-3881/ab5f2e 2019

  4. [4]

    J., & Schlemm, E

    Brockwell, P. J., & Schlemm, E. 2013, Journal of Multivariate Analysis, 115, 217, doi: https://doi.org/10.1016/j.jmva.2012.09.004

    work page doi:10.1016/j.jmva.2012.09.004 2013

  5. [5]

    M., Bentz, M

    Cackett, E. M., Bentz, M. C., & Kara, E. 2021, Nature Astronomy, 5, 693, doi: 10.1038/s41550-021-01401-4

    work page doi:10.1038/s41550-021-01401-4 2021

  6. [6]

    2020, Proceedings of the National Academy of Sciences, 117, 30055, doi: 10.1073/pnas.1912789117

    Cranmer, K., Brehmer, J., & Louppe, G. 2020, Proceedings of the National Academy of Sciences, 117, 30055, doi: 10.1073/pnas.1912789117

    doi:10.1073/pnas.1912789117 2020

  7. [7]

    Cressie, N., & Wikle, C. K. 2011, Statistics for Spatio-Temporal Data, Wiley Series in Probability and Statistics (Hoboken, NJ: John Wiley & Sons)

    2011

  8. [8]

    2017, AJ, 154, 220, doi: 10.3847/1538-3881/aa9332

    Foreman-Mackey, D., et al. 2017, AJ, 154, 220, doi: 10.3847/1538-3881/aa9332

    doi:10.3847/1538-3881/aa9332 2017

  9. [9]

    E., Diggle, P., Guttorp, P., & Fuentes, M., eds

    Gelfand, A. E., Diggle, P., Guttorp, P., & Fuentes, M., eds. 2010, Handbook of Spatial Statistics (Boca Raton, FL: CRC Press)

    2010

  10. [10]

    P., et al

    Gibson, N. P., et al. 2012, MNRAS, 419, 2683, doi: 10.1111/j.1365-2966.2011.19915.x

    arXiv 2012

  11. [11]

    B., Jones, D

    Gilbertson, C., Ford, E. B., Jones, D. E., & Stenning, D. C. 2021, Astronomical Journal, 161, 169, doi: 10.3847/1538-3881/abd9b8

    work page doi:10.3847/1538-3881/abd9b8 2021

  12. [12]

    1997, Geostatistics for Natural Resources Evaluation (Oxford University Press)

    Goovaerts, P. 1997, Geostatistics for Natural Resources Evaluation (Oxford University Press)

    1997

  13. [13]

    D., et al

    Haywood, R. D., et al. 2014, MNRAS, 443, 2517, doi: 10.1093/mnras/stu1320

    doi:10.1093/mnras/stu1320 2014

  14. [14]

    J., Datta, A., Finley, A

    Heaton, M. J., Datta, A., Finley, A. O., et al. 2019, Journal of Agricultural, Biological and Environmental Statistics, 24, 398, doi: 10.1007/s13253-018-00348-w

    doi:10.1007/s13253-018-00348-w 2019

  15. [15]

    Hensman, J., Fusi, N., & Lawrence, N. D. 2013, in Proceedings of the Conference on Uncertainty in Artificial Intelligence (UAI)

    2013

  16. [16]

    Hensman, J., Matthews, A. G. d. G., & Ghahramani, Z. 2015, Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, 38, 351

    2015

  17. [17]

    2020, AJ, 160, 265, doi: 10.3847/1538-3881/abc1e2 Ivezi´ c,ˇZ., et al

    Hu, Z., & Tak, H. 2020, AJ, 160, 265, doi: 10.3847/1538-3881/abc1e2 Ivezi´ c,ˇZ., et al. 2019, ApJ, 873, 111, doi: 10.3847/1538-4357/ab042c

    doi:10.3847/1538-3881/abc1e2 2020

  18. [18]

    E., Stenning, D

    Jones, D. E., Stenning, D. C., Ford, E. B., et al. 2022, The Annals of Applied Statistics, 16, 652 , doi: 10.1214/21-AOAS1471 21

    doi:10.1214/21-aoas1471 2022

  19. [19]

    2021, Statistical Science, 36, 124, doi: 10.1214/19-STS755

    Katzfuss, M., & Guinness, J. 2021, Statistical Science, 36, 124, doi: 10.1214/19-STS755

    doi:10.1214/19-sts755 2021

  20. [20]

    C., Bechtold, J., & Siemiginowska, A

    Kelly, B. C., Bechtold, J., & Siemiginowska, A. 2009, The Astrophysical Journal, 698, 895

    2009

  21. [21]

    C., Becker, A

    Kelly, B. C., Becker, A. C., Sobolewska, M., Siemiginowska, A., & Uttley, P. 2014, The Astrophysical Journal, 788, 33, doi: 10.1088/0004-637X/788/1/33

    doi:10.1088/0004-637x/788/1/33 2014

  22. [22]

    C., Sobolewska, M., & Siemiginowska, A

    Kelly, B. C., Sobolewska, M., & Siemiginowska, A. 2011, ApJ, 730, 52, doi: 10.1088/0004-637X/730/1/52

    doi:10.1088/0004-637x/730/1/52 2011

  23. [23]

    I.-H., Johnson, S

    Li, J. I.-H., Johnson, S. D., Avestruz, C., et al. 2024, ApJ, 977, 223, doi: 10.3847/1538-4357/ad900d

    doi:10.3847/1538-4357/ad900d 2024

  24. [24]

    2016, The Astrophysical Journal, 831, 206, doi: 10.3847/0004-637X/831/2/206

    Li, Y.-R., Wang, J.-M., & Bai, J.-M. 2016, The Astrophysical Journal, 831, 206, doi: 10.3847/0004-637X/831/2/206

    doi:10.3847/0004-637x/831/2/206 2016

  25. [25]

    2020, IEEE Transactions on Neural Networks and Learning Systems, 1, doi: 10.1109/TNNLS.2019.2957109

    Liu, H., Ong, Y., Shen, X., & Cai, J. 2020, IEEE Transactions on Neural Networks and Learning Systems, 1, doi: 10.1109/TNNLS.2019.2957109

    arXiv 2020

  26. [26]

    Winter, M. K. 2016, ApJS, 225, 31, doi: 10.3847/0067-0049/225/2/31

    doi:10.3847/0067-0049/225/2/31 2016

  27. [27]

    J., Bassetto, G., et al

    Lueckmann, J.-M., Gon¸ calves, P. J., Bassetto, G., et al. 2021, Proceedings of the National Academy of Sciences, 118, e2102765118, doi: 10.1073/pnas.2102765118

    work page doi:10.1073/pnas.2102765118 2021

  28. [28]

    2010, The Astrophysical Journal, 721, 1014

    MacLeod, C., Ivezi´ c,ˇZ., Kochanek, C., et al. 2010, The Astrophysical Journal, 721, 1014

    2010

  29. [29]

    L., Ivezi´ c,ˇZ., Sesar, B., et al

    MacLeod, C. L., Ivezi´ c,ˇZ., Sesar, B., et al. 2012, The Astrophysical Journal, 753, 106, doi: 10.1088/0004-637x/753/2/106

    doi:10.1088/0004-637x/753/2/106 2012

  30. [30]

    2007, Stochastic Processes and their Applications, 117, 96, doi: 10.1016/j.spa.2006.05.014

    Marquardt, T., & Stelzer, R. 2007, Stochastic Processes and their Applications, 117, 96, doi: 10.1016/j.spa.2006.05.014

    doi:10.1016/j.spa.2006.05.014 2007

  31. [31]

    D., van Dyk, D

    Meyer, A. D., van Dyk, D. A., Tak, H., & Siemiginowska, A. 2023, ApJ, 950, 37, doi: 10.3847/1538-4357/acbea1

    work page doi:10.3847/1538-4357/acbea1 2023

  32. [32]

    J., Mohamed, S., & Lakshminarayanan, B

    Papamakarios, G., Nalisnick, E., Rezende, D. J., Mohamed, S., & Lakshminarayanan, B. 2021, Journal of Machine Learning Research, 22, 1

    2021

  33. [33]

    2019, Proceedings of Machine Learning Research, 89, 837

    Papamakarios, G., Sterratt, D., & Murray, I. 2019, Proceedings of Machine Learning Research, 89, 837

    2019

  34. [34]

    2015, MNRAS, 452, 2269, doi: 10.1093/mnras/stv1428

    Rajpaul, V., Aigrain, S., Osborn, H., Reece, S., & Roberts, S. 2015, MNRAS, 452, 2269, doi: 10.1093/mnras/stv1428

    doi:10.1093/mnras/stv1428 2015

  35. [35]

    E., & Williams, C

    Rasmussen, C. E., & Williams, C. K. I. 2006, Gaussian Processes for Machine Learning, Adaptive Computation and Machine Learning (Cambridge, MA: MIT Press) S¨ arkk¨ a, S. 2013, Bayesian Filtering and Smoothing (Cambridge: Cambridge University Press), doi: 10.1017/CBO9781139344203

    doi:10.1017/cbo9781139344203 2006

  36. [36]

    2012, Bernoulli, 18, 46, doi: 10.3150/10-BEJ329

    Schlemm, E., & Stelzer, R. 2012, Bernoulli, 18, 46, doi: 10.3150/10-BEJ329

    doi:10.3150/10-bej329 2012

  37. [37]

    N., Denney, K

    Shen, Y., Brandt, W. N., Denney, K. D., et al. 2015, The Astrophysical Journal Supplement Series, 216, 4, doi: 10.1088/0067-0049/216/1/4

    doi:10.1088/0067-0049/216/1/4 2015

  38. [38]

    J., Horne, K., et al

    Shen, Y., Grier, C. J., Horne, K., et al. 2024, ApJS, 272, 26, doi: 10.3847/1538-4365/ad3936

    doi:10.3847/1538-4365/ad3936 2024

  39. [39]

    A., et al

    Tak, H., Mandel, K., van Dyk, D. A., et al. 2017, The Annals of Applied Statistics, 11, 1309, doi: 10.1214/17-AOAS1027

    doi:10.1214/17-aoas1027 2017

  40. [40]

    W., Seeger, M., & Jordan, M

    Teh, Y. W., Seeger, M., & Jordan, M. I. 2005, in JMLR Workshop and Conference Proceedings, Vol. R5, Proceedings of the Tenth International Workshop on Artificial Intelligence and Statistics (AISTATS), 333–340

    2005

  41. [41]

    Titsias, M. K. 2009, in Proceedings of Machine Learning

    2009

  42. [42]

    Wilkins, D. R. 2014, A&A Rv, 22, 72, doi: 10.1007/s00159-014-0072-0

    doi:10.1007/s00159-014-0072-0 2014

  43. [43]

    Vecchia, A. V. 1988, Journal of the Royal Statistical Society: Series B, 50, 297, doi: 10.1111/j.2517-6161.1988.tb01729.x

    arXiv 1988

  44. [44]

    G., et al

    York, D. G., et al. 2000, AJ, 120, 1579, doi: 10.1086/301513

    doi:10.1086/301513 2000

  45. [45]

    T., Ruan, J

    Yu, W., Richards, G. T., Ruan, J. J., et al. 2025, ApJ, 992, 130, doi: 10.3847/1538-4357/adfdd2

    doi:10.3847/1538-4357/adfdd2 2025

  46. [46]

    S., Koz lowski, S., & Peterson, B

    Zu, Y., Kochanek, C. S., Koz lowski, S., & Peterson, B. M. 2016, The Astrophysical Journal, 819, 122, doi: 10.3847/0004-637x/819/2/122

    doi:10.3847/0004-637x/819/2/122 2016

  47. [47]

    S., & Peterson, B

    Zu, Y., Kochanek, C. S., & Peterson, B. M. 2011, The Astrophysical Journal, 735, 80, doi: 10.1088/0004-637x/735/2/80 ´Alvarez, M. A., Luengo, D., Titsias, M. K., & Lawrence, N. D. 2010, in Proceedings of Machine Learning

    doi:10.1088/0004-637x/735/2/80 2011

  48. [48]

    9, Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics (AISTATS), ed

    Research, Vol. 9, Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics (AISTATS), ed. Y. W. Teh & M. Titterington (PMLR), 25–32 ´Alvarez, M. A., Luengo, D., Titsias, M. K., & Lawrence, N. D. 2011, Journal of Machine Learning Research, 12, 1459

    2011