Pith. sign in

REVIEW 3 major objections 5 minor 73 references

Modernizing full posterior inference for surrogate modeling of categorical-output simulation experiments

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Fully Bayesian Gaussian-process classification can scale to tens of thousands of training points by pairing elliptical slice sampling with Vecchia's sparse covariance approximation.

desk verdict Useful recombination of Vecchia and ESS for fully Bayesian GP classification, but the headline UQ claim leans on a data-dependent prior scale that needs sharper sensitivity and calibration analysis. read the letter →

arxiv 2501.14946 v1 pith:TAXP2C5R submitted 2025-01-24 stat.CO

Andrew Cooper , Annie S. Booth , Robert B. Gramacy This is my paper · ORCID
classification stat.CO
keywords GaussianprocessclassificationVecchiaapproximationellipticalslicesamplingsurrogatemodelingbinaryblackholesimulationdeepprocessesuncertaintyquantification
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 argues that fully Bayesian Gaussian-process classification, the version that keeps the entire posterior over latent functions instead of optimizing a point estimate, can be made practical for training sets in the tens of thousands. The recipe pairs elliptical slice sampling, which draws the high-dimensional latent Gaussian variables without the slow mixing of older Metropolis samplers, with Vecchia approximation, which replaces dense covariance inversions with sparse Cholesky factors. On the motivating problem of predicting whether binary black holes form in COMPAS simulations, and on several benchmark classifiers, the resulting method gives predictive probabilities with better logarithmic scores than variational-inference baselines, meaning its uncertainty statements are better calibrated. The same machinery extends to a deep version with input-warping layers for nonstationary classification. If the claim holds, practitioners no longer have to trade full Bayesian uncertainty quantification away to survive large surrogate-modeling campaigns.

What carries the argument

The computational load-bearer is the sparse Cholesky factor U of the Vecchia-approximated precision matrix, Σ(X)^{-1} ≈ UU^T with at most m nonzero entries per row; an elliptical slice sampling proposal Z' is then found by solving U^T Z' = a with a ~ N(0,I), reducing each MCMC update from O($n^{3}$) to O($nm^{3}$). This factorization also supplies the Metropolis acceptance for lengthscale hyperparameters and the stacked predictive equations for new inputs. The calibration load-bearer is the epsilon-information prior: the insulation statistic ωmax counts how many nearest neighbors agree with each training label, and $τ^{2}$ is fixed so that the most insulated point can reach probability 1/(1+ωmax) or its complement; the paper sets ε = 0.001 in all examples. The two pieces together are what let the sampler keep the full latent posterior instead of replacing integration with optimization.

What would settle it

On a benchmark with a known latent probability surface, fit the method with ε = 0.001 and with a cross-validated ε, and check whether 95% posterior intervals for p(x) contain the true probabilities at a near-95% rate on held-out inputs; systematic miscalibration that a VI baseline does not share would refute the UQ claim.

Watch

Extended reading notes

Core claim

The central claim is that Vecchia's sparse approximation to the Gaussian-process covariance is not just a computational crutch for regression likelihoods: it is exactly the missing piece that makes elliptical slice sampling competitive for large-scale classification. ESS needs one multivariate normal draw per update of the n latent values, and a Vecchia precision factorization Σ(X)^{-1} = UU^T turns that draw into a sparse triangular solve at O($nm^{3}$) cost for conditioning sets of size m instead of O($n^{3}$). The paper shows, on the COMPAS binary black hole simulations and benchmark problems, that fully Bayesian sampling with this approximation yields out-of-sample probabilities whose log scores beat doubly stochastic variational inference and scalable variational GP baselines, while matching the accuracy of an unapproximated full sampler when that sampler is still feasible. For nonstationary surfaces, the framework is layered with input-warping GPs to form a deep GP classifier with the same Vecchia/ESS posterior sampling.

Load-bearing premise

The load-bearing premise is that the fixed tuning constant ε = 0.001 in the insulation-based rule for the latent prior scale transfers across problems; if it sets $τ^{2}$ too large or too small, the posterior probabilities come out overconfident or washed out and the claimed UQ advantage over variational inference disappears.

Editorial extensions

If this is right

  • Fully Bayesian GPC, with posterior samples over both latent values and lengthscales, is now feasible for n around 10,000, where previously only point-estimate or variational methods were practical.
  • On the binary black hole formation simulator, practitioners get the highest median classification rate and the best log score among compared methods, with less variability across replicate training sets than VI baselines.
  • For nonstationary classification problems, adding one warping layer (a deep GP classifier) sharpens decision boundaries and improves accuracy and log score, especially at smaller training sizes.
  • The same machinery extends directly to multiclass problems by stacking K-1 independent GP layers and using a generalized logistic link.
  • Surrogate-modeling workflows that need well-calibrated probabilities, such as active learning, sensitivity analysis, and calibration, can use full Bayesian UQ without a large computational penalty.

Reading between the lines

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

  • Inference: because ESS accepts any log-likelihood, the Vecchia/ESS pairing should transfer to Poisson, binomial, or other non-Gaussian likelihoods, suggesting a general full-Bayesian latent-Gaussian-model-at-scale template that the paper does not spell out.
  • Inference: the ε = 0.001 scale heuristic is data-dependent through ωmax, so an automatic cross-validated or fully hierarchical ε may be needed before the method is dropped into unknown physical simulators without tuning.
  • Inference: a head-to-head comparison with the NA-as-zero deep GP emulator for COMPAS chirp mass would test whether classification-then-regression or joint modeling gives better calibrated black hole predictions; the paper lists this as future work, and the machinery here makes such a comparison feasible.
  • Inference: batched parallel ESS chains over many cores or GPUs could push the method toward n = 100,000, because the sparse triangular solve is the only per-update bottleneck.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a fully Bayesian Gaussian process classification (GPC) framework that combines elliptical slice sampling (ESS) with Vecchia approximation, making fully Bayesian latent-GP inference feasible for n in the tens of thousands. The authors also extend the framework to a two-layer deep GP classifier with input warping. They compare their method against doubly stochastic variational inference (DSVI) and scalable variational GP (SVGP) on simulated benchmarks and on a binary black hole (BBH) formation simulator, reporting classification rate and log score. The central computational claim is that Vecchia's sparse precision factorization is ideally suited to ESS because ESS requires only MVN proposals, and the empirical claim is that the resulting sampler provides better-calibrated uncertainty quantification than VI-based alternatives.

Significance. If the claims hold, the paper makes a useful contribution by reviving a sampling-based, fully Bayesian alternative to variational/inducing-point GPC at scale, and by extending it to deep input warping. Strengths include repeated Monte Carlo evaluation over ten instances, use of a proper scoring rule, an open-source R implementation, and a challenging real motivating problem (BBH formation). The computational core, Vecchia-accelerated ESS, is sound and practically valuable. However, the headline UQ-superiority claim rests on a data-dependent prior scale chosen through an insulation heuristic with a fixed tuning constant, and the paper reports no calibration diagnostics beyond log score. The empirical case is therefore not yet as strong as the abstract implies.

major comments (3)
  1. [Section 4.1, Eq. (10), Figure 5] The latent scale tau^2 is set from the training labels through the insulation statistic omega_max with a fixed epsilon = 0.001. Since omega_max is computed from Y, this is an empirical-Bayes use of the response; the resulting posterior is not a coherent update from a prior fixed before seeing Y. The claim that this 'crucially avoids double-dipping from a Bayesian statistical perspective' is thus misleading: it avoids additional MCMC sampling, but not double use of the training labels. Figure 5 shows that the predictive surface changes materially with epsilon, and epsilon = 0.001 is justified only by the determinism of the simulator, with no held-out calibration check across the benchmarks. Because the headline comparison is log score, the paper needs either a sensitivity analysis over epsilon across all datasets, a reliability/calibration analysis, or a hierarchical prior on tau^2. Without this, the claimed UQ advantage over VI is not established.
  2. [Sections 4.2-4.3 and Section 5] The empirical evidence for better uncertainty quantification consists solely of classification rate and log score, defined in Eq. (9). Log score is strictly proper, but it rewards sharpness as well as calibration, so a miscalibrated but overconfident predictive distribution can receive a good log score. The paper reports no reliability diagrams, calibration slope, or expected calibration error for any method or dataset, and the deep extension in Section 5 is evaluated under the same two metrics. Given that the paper's central claim is better-calibrated UQ than VI alternatives, calibration diagnostics should be added before that claim can be accepted.
  3. [Abstract, Section 4.1, Section 5] The phrase 'full posterior inference' overstates the inferential status of the model. tau^2 is fixed at a data-dependent value rather than integrated over, and the deep architecture (number of layers, warping dimension) is fixed in advance. The manuscript should explicitly state that posterior sampling covers the latent Z, lengthscales, and (for DGPC) the warping layer W, conditional on a data-informed prior scale, and it should discuss the sensitivity of the conclusions to that choice. This is a framing issue, but it directly affects the central claim and should be corrected.
minor comments (5)
  1. [Section 1 and Appendix A] There are typos: 'appoach' should be 'approach' in the last paragraph of Section 1, and 'outperms' should be 'outperforms' in Appendix A.
  2. [Equation (6)] The superscript X in U_ji^X is never defined; please clarify that it denotes dependence on the input matrix or remove it for consistency with the rest of the paper.
  3. [Figure 10] SVGP's log scores are described as out-of-frame, but no numerical summaries or inset are provided; please report the actual values so that the magnitude of the difference can be assessed.
  4. [Figure 5] The '+' and 'x' markers for minimum and maximum insulation are not explained in the caption; add a legend or describe them in the text.
  5. [Section 3.3] The statement that the authors are 'the first to recognize its value in the context of GPC via ESS' should be softened or verified against Zilber and Katzfuss (2021) and Cao et al. (2023), which also use Vecchia for latent Gaussian models.

Circularity Check

0 steps flagged · score 2.0 of 10

No formal circularity: derivations are self-contained and externally benchmarked; the label-dependent τ² prior is a tuning concern, not a construction-level equivalence.

full rationale

The paper's derivation chain is self-contained: Eq. (5) defines predictive probabilities by ordinary Bayesian marginalization over the latent GP, Algorithm 1 (ESS) and the Vecchia precision factorization in Eq. (6) are standard computational devices fully specified in the paper, and Algorithm 2 alternates between a Metropolis update for θ and an ESS update for Z, with predictive moments in Eq. (8) averaged over posterior samples. No equation identifies the predicted probabilities with the training labels or with a fitted parameter by construction. The only label-dependent input is the empirical prior scale τ² in Section 4.1, set from the insulation statistic ωmax of Eq. (10) with fixed ε=0.001; this is a data-dependent prior (an empirical-Bayes style double use of Y) and a legitimate calibration concern, but it is not equivalent to the central comparative claim and does not force the log-score comparisons, which are evaluated out-of-sample against external GPflux/GPflow implementations. Citations to Sauer et al. are frequent, but the Vecchia/ESS machinery is restated in the paper, no uniqueness theorem is imported from those papers, and the benchmarks are external. I therefore find no formal circularity; the score reflects the mild label-dependent prior and the density of self-citation, not a construction-level equivalence.

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

The method introduces no new entities; the insulation statistic is a data summary. The free parameters are a hand set tuning constant, a fixed conditioning size, and a data-derived latent scale.

free parameters (3)
  • epsilon (epsilon) = 0.001
    Tuning constant for the epsilon-information prior on tau^2; Section 4.1. Fixed across all examples; predictions are sensitive to its value (Figure 5).
  • Vecchia conditioning set size m = 25
    Section 3.3; number of neighbors used per conditional, chosen following Sauer et al. (2022a), no sensitivity analysis.
  • latent scale tau^2 = set from data via omega_max and epsilon=0.001
    Not integrated out; set per dataset using training labels, which is a form of empirical Bayes (Section 4.1).
assumptions (4)
  • domain assumption A Vecchia approximation with conditioning set size m=25 faithfully represents the true GP prior for ESS proposals and for predictive kriging.
    Section 3.3; m=25 is carried over from Sauer et al. (2022a) without a sensitivity analysis in this paper.
  • ad hoc to paper The insulation statistic omega_i and fixed epsilon=0.001 determine a sensible latent scale tau^2 for every dataset.
    Section 4.1, Eq. (10); Figure 5 shows predictive surfaces depend strongly on epsilon, yet epsilon is fixed without validation.
  • domain assumption A stationary GP with Matern 5/2 kernel on the latent scale adequately captures the simulator response after warping (for DGPC).
    Section 5; the method relies on the warping layer making the latent surface stationary.
  • standard math Standard MVN conditioning (kriging equations) holds under the Vecchia-modified precision matrix for prediction.
    Section 2.1 Eq. (3) and Section 3.4 Eq. (7); inherited from Katzfuss and Guinness (2021).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Modernizing full posterior inference for surrogate modeling of categorical-output simulation experiments." pith.science (2026). https://pith.science/paper/TAXP2C5R

@misc{pith2026250114946,
  author       = {Pith},
  title        = {Pith review of: Modernizing full posterior inference for surrogate modeling of categorical-output simulation experiments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TAXP2C5R}},
  note         = {Machine review of arXiv:2501.14946}
}
read the original abstract

Gaussian processes (GPs) are powerful tools for nonlinear classification in which latent GPs are combined with link functions. But GPs do not scale well to large training data. This is compounded for classification where the latent GPs require Markov chain Monte Carlo integration. Consequently, fully Bayesian, sampling-based approaches had been largely abandoned. Instead, maximization-based alternatives, such as Laplace/variational inference (VI) combined with low rank approximations, are preferred. Though feasible for large training data sets, such schemes sacrifice uncertainty quantification and modeling fidelity, two aspects that are important to our work on surrogate modeling of computer simulation experiments. Here we are motivated by a large scale simulation of binary black hole (BBH) formation. We propose an alternative GP classification framework which uses elliptical slice sampling for Bayesian posterior integration and Vecchia approximation for computational thrift. We demonstrate superiority over VI-based alternatives for BBH simulations and other benchmark classification problems. We then extend our setup to warped inputs for "deep" nonstationary classification.

Figures

Figures reproduced from arXiv: 2501.14946 by the authors.

Figure 1
Figure 1. GP predictions from regression and classification models. Real-valued response data [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. ESS GPC illustration. Top-left: training data with “top-hat” response. Top-right: samples of Z (1:T) after burn-in and thinning. Bottom-left: chain of z (t) values at x1, x25 and x35. Bottom-right: autocorrelation distribution (median and 95% quantiles) for all Z (1:T) . 3.3 Vecchia approximation At its core, Vecchia (1988)’s idea for approximating MVN likelihoods is simple. Any joint probability distribution may be… view at source ↗
Figure 3
Figure 3. Left: 2d “box” example. Middle: run-times for “full” and Vecchia-approximated (m = 25) GPC training. Right: classification accuracy from various GPCs. Algorithm 2: Gibbs sampling procedure for GPC estimation. Initialize θ (1), Z(1) for t = 2, . . . , T do θ (t) ∼ π(θ | X, Z(t−1)) // MH via L(Z | X, θ) Z (t) ∼ π(Z | X, Y, θ(t) ) // ESS via L(Z | X, θ), L(Y | Z) Posterior sampling of Z involves Vecchia-approximated MV… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Posterior mean (left) and variance (right) of py(X ) for the 2d “box” example [ [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Left: Posterior mean µ(x) for various ϵ values specifying τ 2 on the 1d sin example [ [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Top-left: Schaffer no. 4 function on X ∈ [−2, 2]2 . Top-right: Binarized version colored by class membership (black = 0, green = 1), and scaled to input space X ∈ [0, 1]2 . Bottom: Vecchia-GPC posterior mean predictions with n = 500 (left) and n = 10,000 (right) LHS tr…
Figure 7
Figure 7. Figure 7: Schaffer function no. 4 results. in [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: “G” function results. only achieves higher LS when n = 10, 000, and seems to perform inconsistently with less data. Our Vecchia-approximated approach doesn’t perform appreciably different than our full GPC for smaller n. 4.3 Binary black hole formation example The COMP…
Figure 9
Figure 9. Figure 9: COMPAS simulations projected onto two pairs of inputs. Green dots indicate BBH formations. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: BBH comparison. SVGP’s LSs are out-of-frame, being well below the other methods. [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: 2d Schaffer function (left) with slices along x2 = 0.1 (middle, red) and x2 = 0.7 (right, blue). Booth et al. (2024). One increasingly popular strategy was formerly known as input warping (Sampson and Guttorp, 1992; Schmidt and O’Hagan, 2003). The idea is to adjust th…
Figure 12
Figure 12. Figure 12: Left: Diagram of two-layer DGPC. Each arrow corresponds to a separate GP. Right: DGPC model hierarchy, color-coded to match components of the diagram. et al., 2022b). Specifically, take Wj ∼ Nn  0, Σθwj (X)  , then convolve with W (t) j via a random angle γ to propo…
Figure 13
Figure 13. Figure 13: Vecchia GPC (left) and DGPC (right) on 2d Schaffer no. 4 with the identical training points. right panel shows a two-layer DGPC on the same data. Notice how the DGPC provides sharper transitions between labels, especially at the edges/corners. Moreover, it furnishes m…
Figure 14
Figure 14. Figure 14: Two-dimensional “box” results. Some negative outliers for DSVI’s log score are cut off from [PITH_FULL_IMAGE:figures/full_fig_p027_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

73 extracted references · 67 canonical work pages

  1. [1]

    Abrahamsen, P. (1997). A review of G aussian random fields and correlation functions. Norsk Regnesentral/Norwegian Computing Center Oslo, https://www.nr.no/directdownload/917_Rapport.pdf

    work page 1997

  2. [2]

    Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American statistical Association\/ , 88, 422, 669--679

    work page 1993

  3. [3]

    B., and Tokdar, S

    Banerjee, A., Dunson, D. B., and Tokdar, S. T. (2013). Efficient Gaussian process regression for large datasets. Biometrika\/ , 100, 1, 75--89

    work page 2013

  4. [4]

    Bates, D., Maechler, M., Jagan, M., et al. (2010). Matrix: Sparse and dense matrix classes and methods. R package version 0.999375-43, URL http://cran. r-project. org/package= Matrix\/

    work page 2010

  5. [5]

    B., and Ludkovski, M

    Binois, M., Gramacy, R. B., and Ludkovski, M. (2018). Practical heteroscedastic Gaussian process modeling for large simulation experiments. Journal of Computational and Graphical Statistics\/ , 27, 4, 808--821

    work page 2018

  6. [6]

    S., Cooper, A., and Gramacy, R

    Booth, A. S., Cooper, A., and Gramacy, R. B. (2024). Nonstationary G aussian process surrogates. arXiv preprint arXiv:2305.19242\/

    arXiv 2024

  7. [7]

    and Gramacy, R

    Broderick, T. and Gramacy, R. B. (2011). Classification and categorical inputs with treed Gaussian process models. Journal of classification\/ , 28, 2, 244--270

    work page 2011

  8. [8]

    Cao, J., Kang, M., Jimenez, F., Sang, H., Schäfer, F., and Katzfuss, M. (2023). Variational Sparse Inverse Cholesky Approximation for Latent Gaussian Processes via Double Kullback-Leibler Minimization

    work page 2023

Show all 73 references
  1. [9]

    A., Christianson, R

    Cole, D. A., Christianson, R. B., and Gramacy, R. B. (2021). Locally induced Gaussian processes for large-scale simulation experiments. Statistics and Computing\/ , 31, 3, 1--21

    2021

  2. [10]

    Damianou, A. C. and Lawrence, N. D. (2012). Deep Gaussian Processes. Proceedings of Machine Learning Research\/

    2012

  3. [11]

    C., Titsias, M

    Damianou, A. C., Titsias, M. K., Gr, M., Lawrence, N. D., and Storkey, A. (2016). Variational Inference for Latent Variables and Uncertain Inputs in Gaussian Processes

    2016

  4. [12]

    Datta, A. (2021). Sparse Cholesky matrices in spatial statistics. arXiv preprint arXiv:2102.13299\/

    2021 arXiv

  5. [13]

    O., and Gelfand, A

    Datta, A., Banerjee, S., Finley, A. O., and Gelfand, A. E. (2016). Hierarchical nearest-neighbor Gaussian process models for large geostatistical datasets. Journal of the American Statistical Association\/ , 111, 514, 800--812

    2016

  6. [14]

    M., Girolami, M

    Dunlop, M. M., Girolami, M. A., Stuart, A. M., and Teckentrup, A. L. (2018). How deep are deep Gaussian processes? Journal of Machine Learning Research\/ , 19, 54, 1--46

    2018

  7. [15]

    P., Hensman, J., and John, S

    Dutordoir, V., Salimbeni, H., Hambro, E., McLeod, J., Leibfried, F., Artemev, A., van der Wilk, M., Deisenroth, M. P., Hensman, J., and John, S. (2021). GPflux: A library for Deep Gaussian Processes. arXiv:2104.05674\/

    2021 arXiv

  8. [16]

    and Sanderson, C

    Eddelbuettel, D. and Sanderson, C. (2014). RcppArmadillo: Accelerating R with high-performance C++ linear algebra. Computational statistics & data analysis\/ , 71, 1054--1063

    2014

  9. [17]

    Emery, X. (2009). The kriging update equations and their application to the selection of neighboring data. Computational Geosciences\/ , 13, 3, 269--280

    2009

  10. [18]

    u hwirth-Schnatter, S. and Fr \

    Fr \"u hwirth-Schnatter, S. and Fr \"u hwirth, R. (2010). Data augmentation and MCMC for binary and multinomial logit models. Statistical modelling and regression structures: Festschrift in honour of Ludwig Fahrmeir\/ , 111--132

    2010

  11. [19]

    B., Stern, H

    Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian data analysis\/ . Chapman and Hall/CRC

    1995

  12. [20]

    and Raftery, A

    Gneiting, T. and Raftery, A. (2007). Strictly proper scoring rules, prediction, and estimation. Journal of the American Statistical Association\/ , 102, 477, 359--378

    2007

  13. [21]

    Gramacy, R. B. (2020). Surrogates: G aussian Process Modeling, Design and Optimization for the Applied Sciences\/ . Boca Raton, Florida: Chapman Hall/CRC. http://bobby.gramacy.com/surrogates/

    2020

  14. [22]

    Gramacy, R. B. and Apley, D. W. (2015). Local Gaussian process approximation for large computer experiments. Journal of Computational and Graphical Statistics\/ , 24, 2, 561--578

    2015

  15. [23]

    Gramacy, R. B. and Lee, H. K. (2012). Cases for the nugget in modeling computer experiments. Statistics and Computing\/ , 22, 713--722

    2012

  16. [24]

    Gramacy, R. B. and Polson, N. G. (2011). Particle Learning of Gaussian Process Models for Sequential Design and Optimization. Journal of Computational and Graphical Statistics\/ , 20, 1, 102--118

    2011

  17. [25]

    Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika\/

    1970

  18. [26]

    Hensman, J., Matthews, A., and Ghahramani, Z. (2015). Scalable variational Gaussian process classification. In Artificial Intelligence and Statistics\/ , 351--360. PMLR

    2015

  19. [27]

    Higdon, D. (1998). A process-convolution approach to modelling temperatures in the North Atlantic Ocean. Environmental and Ecological Statistics\/ , 5, 173--190

    1998

  20. [28]

    and Yin, G

    Jin, H. and Yin, G. (2021). Unit information prior for adaptive information borrowing from multiple historical datasets. Statistics in Medicine\/ , 40, 25, 5657--5672

    2021

  21. [29]

    Jones, D., Schonlau, M., and Welch, W. (1998). Efficient global optimization of expensive black-box functions. Journal of Global Optimization\/ , 13, 4, 455--492

    1998

  22. [30]

    and Guinness, J

    Katzfuss, M. and Guinness, J. (2021). A General Framework for Vecchia Approximations of Gaussian Processes. Statistical Science\/ , 36, 124--141

    2021

  23. [31]

    Katzfuss, M., Guinness, J., Gong, W., and Zilber, D. (2020). Vecchia approximations of Gaussian-process predictions. Journal of Agricultural, Biological and Environmental Statistics\/ , 25, 383--414

    2020

  24. [32]

    Katzfuss, M., Guinness, J., and Lawrence, E. (2022). Scaled Vecchia approximation for fast computer-model emulation. SIAM/ASA Journal on Uncertainty Quantification\/ , 10, 2, 537--554

    2022

  25. [33]

    and O'Hagan, A

    Kennedy, M. and O'Hagan, A. (2001). Bayesian calibration of computer models. Journal of the Royal Statistical Society: Series B (Statistical Methodology)\/ , 63, 3, 425--464

    2001

  26. [34]

    Kingma, D. P. and Welling, M. (2022). Auto-Encoding Variational Bayes

    2022

  27. [35]

    E., and Figueiras-Vidal, A

    L \'a zaro-Gredilla, M., Quinonero-Candela, J., Rasmussen, C. E., and Figueiras-Vidal, A. R. (2010). Sparse spectrum Gaussian process regression. The Journal of Machine Learning Research\/ , 11, 1865--1881

    2010

  28. [36]

    Lin, L., Bingham, D., Broekgaarden, F., and Mandel, I. (2021). Uncertainty Quantification of a Computer Model for Binary Black Hole Formation. Annals of Applied Statistics\/ . Real world example of gp's for classification for purpose of classifying types of black holes

    2021

  29. [37]

    Marrel, A., Iooss, B., Laurent, B., and Roustant, O. (2009). Calculations of S obol indices for the G aussian process metamodel. Reliability Engineering & System Safety\/ , 94, 3, 742--751

    2009

  30. [38]

    Matheron, G. (1963). Principles of geostatistics. Economic geology\/ , 58, 8, 1246--1266

    1963

  31. [39]

    Matthews, A. G. d. G., van der Wilk , M., Nickson, T., Fujii, K., Boukouvalas , A., Le \'o n-Villagr \'a , P., Ghahramani, Z., and Hensman, J. (2017). GP flow: A G aussian process library using T ensor F low . Journal of Machine Learning Research\/ , 18, 40, 1--6

    2017

  32. [40]

    and Nelder, J

    McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models\/ . London: Chapman & Hall / CRC

    1989

  33. [41]

    D., Beckman, R

    McKay, M. D., Beckman, R. J., and Conover, W. J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics\/ , 21, 2, 239--245

    1979

  34. [42]

    and Ramos, F

    Melkumyan, A. and Ramos, F. T. (2009). A sparse covariance function for exact Gaussian process inference in large datasets. In Twenty-first international joint conference on artificial intelligence\/

    2009

  35. [43]

    Minka, T. P. (2001). A family of algorithms for approximate Bayesian inference. Ph.D. thesis, Massachusetts Institute of Technology

    2001

  36. [44]

    D., Mitchell, T

    Morris, M. D., Mitchell, T. J., and Ylvisaker, D. (1993). Bayesian Design and Analysis of Computer Experiments: Use of Derivatives in Surface Prediction. Technometrics\/ , 35, 3, 243--255

    1993

  37. [45]

    Murray, I., Prescott, R., David, A., and Mackay, J. C. (2010). Elliptical slice sampling

    2010

  38. [46]

    Neal, R. (1998). Regression and classification using Gaussian process priors. Bayesian statistics\/ , 6, 475

    1998

  39. [47]

    Neal, R. M. (2003). Slice sampling. The annals of statistics\/ , 31, 3, 705--767

    2003

  40. [48]

    and Rasmussen, C

    Quinonero-Candela, J. and Rasmussen, C. E. (2005). A unifying view of sparse approximate Gaussian process regression. The Journal of Machine Learning Research\/ , 6, 1939--1959

    2005

  41. [49]

    E., and De, C

    Quiñonero, J., Quiñonero-Candela, Q., Rasmussen, C. E., and De, C. M. (2005). A Unifying View of Sparse Approximate Gaussian Process Regression. Inducing points

    2005

  42. [50]

    Rasmussen, C. E. and Williams, C. K. I. (2006). Gaussian processes for machine learning\/ . MIT Press

    2006

  43. [51]

    M., and Lagos, C

    Rauf, L., Howlett, C., Davis, T. M., and Lagos, C. D. (2023). Exploring binary black hole mergers and host galaxies with shark and COMPAS. Monthly Notices of the Royal Astronomical Society\/ , 523, 4, 5719--5737

    2023

  44. [52]

    Remes, S., Heinonen, M., and Kaski, S. (2017). Non-stationary spectral kernels. Advances in neural information processing systems\/ , 30

    2017

  45. [53]

    W., Boyett , K

    Riley , J., Agrawal , P., Barrett , J. W., Boyett , K. N. K., Broekgaarden , F. S., Chattopadhyay , D., Gaebel , S. M., Gittins , F., Hirai , R., Howitt , G., Justham , S., Khandelwal , L., Kummer , F., Lau , M. Y. M., Mandel , I., de Mink , S. E., Neijssel , C., Riley , T., v...

    2022

  46. [54]

    P., Casella, G., and Casella, G

    Robert, C. P., Casella, G., and Casella, G. (1999). Monte Carlo statistical methods\/ , vol. 2. Springer

    1999

  47. [55]

    and Deisenroth, M

    Salimbeni, H. and Deisenroth, M. (2017). Doubly Stochastic Variational Inference for Deep Gaussian Processes. 31st Conference on Neural Information Processing Systems\/ . Dsvi

    2017

  48. [56]

    Sampson, P. D. and Guttorp, P. (1992). Nonparametric estimation of nonstationary spatial covariance structure. Journal of the American Statistical Association\/ , 87, 417, 108--119

    1992

  49. [57]

    Santner, T., Williams, B., and Notz, W. (2018). The Design and Analysis of Computer Experiments, Second Edition\/ . New York, NY: Springer--Verlag

    2018

  50. [58]

    Sauer, A., Cooper, A., and Gramacy, R. B. (2022 a ). Vecchia-approximated deep Gaussian processes for computer experiments. Journal of Computational and Graphical Statistics\/ , 32, 3, 824--837

    2022

  51. [59]

    B., and Higdon, D

    Sauer, A., Gramacy, R. B., and Higdon, D. (2022 b ). Active learning for deep G aussian process surrogates. Technometrics\/ , 65, 1, 4--18

    2022

  52. [60]

    Schmidt, A. M. and O'Hagan, A. (2003). Bayesian inference for non-stationary spatial covariance structure via spatial deformations. Journal of the Royal Statistical Society Series B: Statistical Methodology\/ , 65, 3, 743--758

    2003

  53. [61]

    and Jordan, M

    Seeger, M. and Jordan, M. (2004). Sparse Gaussian process classification with multiple classes. Tech. rep., Department of Statistics, University of Berkeley, CA

    2004

  54. [62]

    and Ghahramani, Z

    Snelson, E. and Ghahramani, Z. (2006). Sparse Gaussian Processes using Pseudo-inputs . Advances in Neural Information Processing Systems 18\/ , 1257--1264

    2006

  55. [63]

    L., Chi, Z., and Welty, L

    Stein, M. L., Chi, Z., and Welty, L. J. (2004). Approximating likelihoods for large spatial data sets. Journal of the Royal Statistical Society Series B: Statistical Methodology\/ , 66, 2, 275--296

    2004

  56. [64]

    R., Stein, M

    Stroud, J. R., Stein, M. L., and Lysen, S. (2017). Bayesian and maximum likelihood estimation for Gaussian processes on an incomplete lattice. Journal of computational and Graphical Statistics\/ , 26, 1, 108--120

    2017

  57. [65]

    and Bingham, D

    Surjanovic, S. and Bingham, D. (2013). Virtual Library of Simulation Experiments: Test Functions and Datasets. Retrieved May 21, 2024, from http://www.sfu.ca/ ssurjano

    2013

  58. [66]

    Titsias, M. (2009). Variational Learning of Inducing Variables in Sparse Gaussian Processes . In Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics\/ , eds. D. van Dyk and M. Welling, vol. 5 of Proceedings of Machine Learning Research\...

    2009

  59. [67]

    Vecchia, A. (1988). Estimation and model identification for continuous spatial processes. Journal of the Royal Statistical Society: Series B (Methodological)\/ , 50, 2, 297--312

    1988

  60. [68]

    Williams, C. K. and Barber, D. (1998). Bayesian classification with Gaussian processes. IEEE Transactions on pattern analysis and machine intelligence\/ , 20, 12, 1342--1351

    1998

  61. [69]

    Wu, L., Pleiss, G., and Cunningham, J. (2022). Variational Nearest Neighbor Gaussian Processes. arXiv preprint arXiv:2202.01694\/

    2022 arXiv

  62. [70]

    Yazdi, F., Bingham, D., and Williamson, D. (2024). Deep Gaussian Process Emulation and Uncertainty Quantification for Large Computer Experiments

    2024

  63. [71]

    and Katzfuss, M

    Zilber, D. and Katzfuss, M. (2021). Vecchia–Laplace approximations of generalized Gaussian processes for big non-Gaussian spatial data. Computational Statistics & Data Analysis\/ , 153, 107081

    2021

  64. [72]

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

    ENTRY address author booktitle chapter edition editor howpublished institution 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 a...

  65. [73]

    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 gl...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.