pith. sign in

arxiv: 2510.08974 · v2 · submitted 2025-10-10 · 📊 stat.CO · cs.NA· math.NA

Bayesian Active Learning for Bayesian Model Updating: the Art of Acquisition Functions and Beyond

Jingwen Song , Pengfei Wei This is my paper

Pith reviewed 2026-05-18 08:25 UTC · model grok-4.3

classification 📊 stat.CO cs.NAmath.NA
keywords Bayesian quadratureacquisition functionsactive learningBayesian model updatingposterior estimationmodel evidencetransitional methodsuncertainty quantification
0
0 comments X

The pith

Four new acquisition functions measure prediction uncertainties to improve Bayesian quadrature in model updating.

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

This paper introduces four novel acquisition functions for Bayesian active learning applied to Bayesian model updating. These functions are crafted to quantify the prediction uncertainty of the posterior, the role of evidence in that uncertainty, and the anticipated decreases in those uncertainties when adding new integration points. The goal is to optimize the selection of points where the expensive model is evaluated, allowing accurate estimation of posteriors and evidences even when the posterior has complicated features like unequal multi-modalities, nonlinear dependencies, and high sharpness. The methods are extended to transitional Bayesian quadrature with additional refinements for better performance.

Core claim

The paper primarily develops four new acquisition functions inspired by distinct intuitions on expected rewards. Mathematically, these functions measure the prediction uncertainty of the posterior, the contribution to prediction uncertainty of the evidence, the expected reduction of prediction uncertainty concerning the posterior, and that concerning the evidence. This provides flexibility for designing integration points. The acquisition functions are extended to the transitional Bayesian quadrature scheme with refinements to achieve high efficiency and robustness for complex posteriors.

What carries the argument

Acquisition functions that govern the active generation of integration points in Bayesian quadrature schemes for estimating posteriors and model evidences.

If this is right

  • Provides flexibility for highly effective design of integration points based on different uncertainty measures.
  • Extends to transitional BQ scheme enabling handling of posteriors with multi-modalities of unequal importance.
  • Refinements ensure high efficiency and robustness on posteriors with nonlinear dependencies and high sharpness.
  • Demonstrated effectiveness through benchmark studies and engineering applications.

Where Pith is reading between the lines

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

  • These acquisition functions might be combined with other active learning strategies to further optimize computational budgets.
  • Applications could extend to high-dimensional problems where traditional quadrature struggles.
  • The approach highlights the value of tailoring acquisition to specific aspects of uncertainty in inference tasks.

Load-bearing premise

Intuitions based on expected rewards for the acquisition functions will result in superior numerical performance and robustness for complex posteriors without introducing instabilities or biases.

What would settle it

Numerical experiments on a posterior with sharp multi-modal features where the new methods show higher error rates or require more model evaluations than established acquisition functions.

Figures

Figures reproduced from arXiv: 2510.08974 by Jingwen Song, Pengfei Wei.

Figure 1
Figure 1. Figure 1: Reference results of unnormalized posteriors and the associated model evidences of the four 2D examples . [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Results of the posteriors defined by U1, which are estimated by BQ algorithm driven by the four acquisition functions, together with the training points and the mean estimate of model evidence. Next, the results for the problem formulated with U3, as reported in [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Results of the posteriors defined by U2, generated with the BQ algorithm. As observed in [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Results of the posteriors defined by U3, generated with the BQ algorithm [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Results of the posteriors of the forth 2D example, generated with the BQ algorithm. [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results for posterior defined with U1, generated by the TBQ algorithm driven by PUQ (1st row), PVC (2nd row), PLUR (3nd row) and PEUR (4th row). The newly added training points generated each stage are marked on the corresponding subplots. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Results of the posteriors defined by U2, generated by TBQ equipped with the four proposed acquisition functions. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Results of the posteriors defined by U3, generated by the TBQ algorithm. With identical parameter configurations, and 12 initial training samples randomly generated, the TBQ algorithm equipped with each acquisition function was executed ten times, the results of model evidence are summarized in [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Results of the posteriors defined by U4, generated by the TBQ algorithm. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of results of the pairwise marginal posteriors for the 10-dimensional example. Those displayed in the [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The 3DoF undamped spring-mass system. The algorithm parameters of TBQ are set to be ϵ = 0.02, ς = 1, and NMC = 104 . Initialized with the same 12 training samples, the TBQ algorithm is implemented with each of the four acquisition functions, and resultant posterior samples of the last tempering stage are reported in the first four rows of [PITH_FULL_IMAGE:figures/full_fig_p035_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of posterior results for the 3DoF system. Note that the plotting range is smaller than the support of [PITH_FULL_IMAGE:figures/full_fig_p036_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Battery module cooling model, with (a) indicating the mesh model and (b). being the predicted temperature [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Results of marginal posterior densities for the battery cooling analysis example. [PITH_FULL_IMAGE:figures/full_fig_p038_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Results of bivariate marginal posteriors of the three influential parameters for the battery cooling analysis example. [PITH_FULL_IMAGE:figures/full_fig_p039_15.png] view at source ↗
read the original abstract

Estimating posteriors and the associated model evidences, with desired accuracy and affordable computational cost, is a core issue of Bayesian model updating, and can be of great challenge given expensive-to-evaluate models and posteriors with complex features such as multi-modalities of unequal importance, nonlinear dependencies and high sharpness. Bayesian Quadrature (BQ) equipped with active learning has emerged as a competitive framework for tackling this challenge, as it provides flexible balance between computational cost and accuracy. The performance of a BQ scheme is fundamentally dictated by the acquisition function as it exclusively governs the active generation of integration points. After reexamining one of the most advanced acquisition function from a prospective inference perspective and reformulating the quadrature rules for prediction, four new acquisition functions, inspired by distinct intuitions on expected rewards, are primarily developed, all of which are accompanied by elegant interpretations and highly efficient numerical estimators. Mathematically, these four acquisition functions measure, respectively, the prediction uncertainty of posterior, the contribution to prediction uncertainty of evidence, as well as the expected reduction of prediction uncertainties concerning posterior and evidence, and thus provide flexibility for highly effective design of integration points. These acquisition functions are further extended to the transitional BQ scheme, along with several specific refinements, to tackle the above-mentioned challenges with high efficiency and robustness. Effectiveness of the developments is ultimately demonstrated with extensive benchmark studies and application to an engineering example.

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

1 major / 2 minor

Summary. The paper re-examines an existing acquisition function for Bayesian quadrature (BQ) from a prospective inference viewpoint, reformulates the underlying quadrature rules for prediction, and introduces four new acquisition functions motivated by distinct expected-reward intuitions. These functions respectively quantify posterior predictive uncertainty, the evidence contribution to that uncertainty, and the expected reductions in each; they are accompanied by efficient numerical estimators and extended to a transitional BQ scheme with ad-hoc refinements for robustness. The developments are demonstrated on benchmark problems involving multi-modal, nonlinear, and sharp posteriors as well as an engineering application.

Significance. If the empirical performance gains hold under the stated conditions, the new acquisition functions would supply a principled and flexible toolkit for active point selection in BQ-based Bayesian model updating, potentially improving the trade-off between computational cost and accuracy for expensive forward models with complex posterior features. The explicit interpretations of the acquisition functions also contribute to the conceptual understanding of uncertainty reduction in this setting.

major comments (1)
  1. [Benchmark studies and transitional BQ extension] The central claim that the four new acquisition functions deliver 'high efficiency and robustness' on posteriors with multi-modalities of unequal importance, nonlinear dependencies, and high sharpness rests on empirical benchmark studies alone. No convergence rates, bias bounds, or worst-case stability analysis for the resulting quadrature estimators are supplied, leaving the translation from expected-reward intuition to numerical reliability unverified (see the description of the benchmark studies and the transitional BQ extension).
minor comments (2)
  1. Notation for the four new acquisition functions and their estimators should be introduced with explicit definitions and cross-references to the reformulated quadrature rules to improve readability.
  2. The abstract and introduction would benefit from a concise table or diagram contrasting the existing acquisition function with the four new ones in terms of the uncertainty quantities they target.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and positive evaluation of the paper's significance. Below we respond to the major comment and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Benchmark studies and transitional BQ extension] The central claim that the four new acquisition functions deliver 'high efficiency and robustness' on posteriors with multi-modalities of unequal importance, nonlinear dependencies, and high sharpness rests on empirical benchmark studies alone. No convergence rates, bias bounds, or worst-case stability analysis for the resulting quadrature estimators are supplied, leaving the translation from expected-reward intuition to numerical reliability unverified (see the description of the benchmark studies and the transitional BQ extension).

    Authors: We agree that the claims of efficiency and robustness for the proposed acquisition functions rest on empirical validation. The functions are derived from distinct expected-reward intuitions, each accompanied by efficient numerical estimators, and their performance is demonstrated across benchmark problems that specifically include multi-modal posteriors of unequal importance, nonlinear dependencies, and high sharpness, as well as within the transitional BQ scheme with the described refinements. We do not supply convergence rates, bias bounds, or worst-case stability analysis, as obtaining such theoretical guarantees for these novel acquisition functions under general posterior conditions is technically challenging and lies outside the primary scope of the manuscript, which emphasizes method development and practical applicability. To address the referee's concern, we will revise the manuscript by (i) explicitly qualifying the claims to reflect the empirical nature of the supporting evidence and (ii) adding a short discussion paragraph on the current verification approach and the value of future theoretical work. This constitutes a partial revision that improves transparency while preserving the core contributions. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in acquisition function derivations

full rationale

The paper proposes four new acquisition functions motivated by separate intuitions on expected rewards for reducing posterior and evidence prediction uncertainties, with explicit mathematical definitions and numerical estimators provided independently of any fitted parameters or prior results. These are extended to transitional BQ with refinements, and effectiveness is shown via benchmark studies and an engineering application rather than by algebraic reduction to inputs. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the described chain; the derivations remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard assumptions from Bayesian quadrature and active learning literature for approximating integrals in posterior and evidence estimation. No explicit free parameters, new axioms, or invented entities are described in the abstract; the contribution centers on new acquisition functions derived from expected-reward intuitions.

axioms (1)
  • domain assumption Bayesian quadrature with active learning can flexibly balance computational cost and accuracy for estimating posteriors and model evidences in complex settings.
    This is the core framework invoked throughout the abstract for tackling expensive models and complex posterior features.

pith-pipeline@v0.9.0 · 5779 in / 1290 out tokens · 46687 ms · 2026-05-18T08:25:04.395630+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    M. C. Kennedy, A. O’Hagan, Bayesian calibration of computer models, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63 (3) (2001) 425–464

    work page 2001

  2. [2]

    S. Bi, M. Beer, S. Cogan, J. Mottershead, Stochastic model updating with uncertainty quantification: an overview and tutorial, Mechanical Systems and Signal Processing 204 (2023) 110784

    work page 2023

  3. [3]

    Huang, C

    Y. Huang, C. Shao, B. Wu, J. L. Beck, H. Li, State-of-the-art review on Bayesian inference in structural system identification and damage assessment, Advances in Structural Engineering 22 (6) (2019) 1329– 1351

    work page 2019

  4. [4]

    Y. Zeng, J. Zeng, M. D. Todd, Z. Hu, Data augmentation based on image translation for bayesian inference-based damage diagnostics of miter gates, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering 11 (1) (2025) 011103

    work page 2025

  5. [5]

    T. Yin, A practical bayesian framework for structural model updating and prediction, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering 8 (1) (2022) 04021073

    work page 2022

  6. [6]

    A. F. Psaros, X. Meng, Z. Zou, L. Guo, G. E. Karniadakis, Uncertainty quantification in scientific machine learning: Methods, metrics, and comparisons, Journal of Computational Physics 477 (2023) 111902

    work page 2023

  7. [7]

    Jerez, H

    D. Jerez, H. Jensen, M. Beer, M. Broggi, Contaminant source identification in water distribution networks: A Bayesian framework, Mechanical Systems and Signal Processing 159 (2021) 107834

    work page 2021

  8. [8]

    Bingham, T

    D. Bingham, T. Butler, D. Estep, Inverse problems for physics-based process models, Annual Review of Statistics and Its Application 11 (2024)

    work page 2024

  9. [9]

    Bryutkin, M

    A. Bryutkin, M. E. Levine, I. Urteaga, Y. Marzouk, Canonical bayesian linear system identification, arXiv preprint arXiv:2507.11535 (2025)

    work page arXiv 2025

  10. [10]

    Van Ravenzwaaij, P

    D. Van Ravenzwaaij, P. Cassey, S. D. Brown, A simple introduction to Markov Chain Monte–Carlo sampling, Psychonomic Bulletin & Review 25 (1) (2018) 143–154

    work page 2018

  11. [11]

    G. L. Jones, Q. Qin, Markov chain Monte Carlo in practice, Annual Review of Statistics and Its Application 9 (1) (2022) 557–578

    work page 2022

  12. [12]

    Metropolis, A

    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, Equation of state calcu- lations by fast computing machines, The Journal of Chemical Physics 21 (6) (1953) 1087–1092

    work page 1953

  13. [13]

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

    work page 1970

  14. [14]

    R. M. Neal, et al., MCMC using Hamiltonian dynamics, Handbook of markov chain monte carlo 2 (11) (2011) 2

    work page 2011

  15. [15]

    M.Betancourt, S.Byrne, S.Livingstone, M.Girolami, ThegeometricfoundationsofHamiltonianMonte Carlo, Bernoulli (2017) 2257–2298. 44

    work page 2017

  16. [16]

    Ching, Y.-C

    J. Ching, Y.-C. Chen, Transitional Markov Chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging, Journal of Engineering Mechanics 133 (7) (2007) 816–832

    work page 2007

  17. [17]

    W. Betz, I. Papaioannou, D. Straub, Transitional Markov Chain Monte Carlo: observations and im- provements, Journal of Engineering Mechanics 142 (5) (2016) 04016016

    work page 2016

  18. [18]

    S. L. Cotter, G. O. Roberts, A. M. Stuart, D. White, Mcmc methods for functions: modifying old algorithms to make them faster, Statistical Science 28 (3) (2013) 424–446

    work page 2013

  19. [19]

    Hairer, A

    M. Hairer, A. M. Stuart, S. J. Vollmer, Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions, The Annals of Applied Probability 24 (6) (2014) 2455–2490

    work page 2014

  20. [20]

    S. Syed, A. Bouchard-Côté, G. Deligiannidis, A. Doucet, Non-reversible parallel tempering: a scalable highlyparallelMCMCscheme, JournaloftheRoyalStatisticalSocietySeriesB:StatisticalMethodology 84 (2) (2022) 321–350

    work page 2022

  21. [21]

    Roy, Convergence diagnostics for Markov Chain Monte Carlo, Annual Review of Statistics and Its Application 7 (1) (2020) 387–412

    V. Roy, Convergence diagnostics for Markov Chain Monte Carlo, Annual Review of Statistics and Its Application 7 (1) (2020) 387–412

    work page 2020

  22. [22]

    L. F. South, M. Riabiz, O. Teymur, C. J. Oates, Postprocessing of MCMC, Annual Review of Statistics and Its Application 9 (1) (2022) 529–555

    work page 2022

  23. [23]

    D. G. Tzikas, A. C. Likas, N. P. Galatsanos, The variational approximation for Bayesian inference, IEEE Signal Processing Magazine 25 (6) (2008) 131–146

    work page 2008

  24. [24]

    Zhang, J

    C. Zhang, J. Bütepage, H. Kjellström, S. Mandt, Advances in variational inference, IEEE transactions on pattern analysis and machine intelligence 41 (8) (2019) 2008–2026

    work page 2019

  25. [25]

    C. W. Fox, S. J. Roberts, A tutorial on variational Bayesian inference, Artificial intelligence review 38 (2012) 85–95

    work page 2012

  26. [26]

    arXiv preprint arXiv:2103.01327 , year=

    M.-N. Tran, T.-N. Nguyen, V.-H. Dao, A practical tutorial on variational bayes, arXiv preprint arXiv:2103.01327 (2021)

    work page arXiv 2021

  27. [27]

    Opper, D

    M. Opper, D. Saad, Advanced mean field methods: Theory and practice, MIT press, 2001

    work page 2001

  28. [28]

    Ranganath, S

    R. Ranganath, S. Gerrish, D. Blei, Black box variational inference, in: Artificial intelligence and statis- tics, PMLR, 2014, pp. 814–822

    work page 2014

  29. [29]

    Kucukelbir, D

    A. Kucukelbir, D. Tran, R. Ranganath, A. Gelman, D. M. Blei, Automatic differentiation variational inference, Journal of machine learning research 18 (14) (2017) 1–45

    work page 2017

  30. [30]

    Rezende, S

    D. Rezende, S. Mohamed, Variational inference with normalizing flows, in: International conference on machine learning, PMLR, 2015, pp. 1530–1538

    work page 2015

  31. [31]

    F. Hong, P. Wei, S. Bi, M. Beer, Efficient variational Bayesian model updating by Bayesian active learning, Mechanical Systems and Signal Processing 224 (2025) 112113

    work page 2025

  32. [32]

    P. Ni, J. Li, H. Hao, Q. Han, X. Du, Probabilistic model updating via variational Bayesian inference and adaptive Gaussian process modeling, Computer Methods in Applied Mechanics and Engineering 383 (2021) 113915

    work page 2021

  33. [33]

    Yoshida, T

    I. Yoshida, T. Nakamura, S.-K. Au, Bayesian updating of model parameters using adaptive Gaussian process regression and particle filter, Structural Safety 102 (2023) 102328

    work page 2023

  34. [34]

    R.Baptista, Y.Marzouk, O.Zahm, Ontherepresentationandlearningofmonotonetriangulartransport maps, Foundations of Computational Mathematics 24 (6) (2024) 2063–2108. 45

    work page 2024

  35. [35]

    Papamakarios, E

    G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mohamed, B. Lakshminarayanan, Normalizing flows for probabilistic modeling and inference, Journal of Machine Learning Research 22 (57) (2021) 1–64

    work page 2021

  36. [36]

    Hennig, M

    P. Hennig, M. A. Osborne, H. P. Kersting, Probabilistic Numerics: Computation as Machine Learning, Cambridge University Press, 2022

    work page 2022

  37. [37]

    Osborne, R

    M. Osborne, R. Garnett, Z. Ghahramani, D. K. Duvenaud, S. J. Roberts, C. Rasmussen, Active learning of model evidence using Bayesian quadrature, Advances in neural information processing systems 25 (2012)

    work page 2012

  38. [38]

    Gunter, M

    T. Gunter, M. A. Osborne, R. Garnett, P. Hennig, S. J. Roberts, Sampling for inference in probabilistic models with fast Bayesian quadrature, Advances in neural information processing systems 27 (2014)

    work page 2014

  39. [39]

    J. Song, Z. Liang, P. Wei, M. Beer, Sampling-based adaptive Bayesian quadrature for probabilistic model updating, Computer Methods in Applied Mechanics and Engineering 433 (2025) 117467

    work page 2025

  40. [40]

    Kitahara, T

    M. Kitahara, T. Kitahara, Sequential and adaptive probabilistic integration for Bayesian model updat- ing, Mechanical Systems and Signal Processing 223 (2025) 111825

    work page 2025

  41. [41]

    Wei, Bayesian model inference with complex posteriors: Exponential-impact-informed bayesian quadrature, Mechanical Systems and Signal Processing (2025)

    P. Wei, Bayesian model inference with complex posteriors: Exponential-impact-informed bayesian quadrature, Mechanical Systems and Signal Processing (2025)

    work page 2025

  42. [42]

    Acerbi, An exploration of acquisition and mean functions in Variational Bayesian Monte Carlo, in: Symposium on Advances in Approximate Bayesian Inference, PMLR, 2019, pp

    L. Acerbi, An exploration of acquisition and mean functions in Variational Bayesian Monte Carlo, in: Symposium on Advances in Approximate Bayesian Inference, PMLR, 2019, pp. 1–10

    work page 2019

  43. [43]

    Chopin, T

    N. Chopin, T. Lelièvre, G. Stoltz, Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors, Statistics and Computing 22 (2012) 897–916

    work page 2012

  44. [44]

    Friston, L

    K. Friston, L. Da Costa, N. Sajid, C. Heins, K. Ueltzhöffer, G. A. Pavliotis, T. Parr, The free energy principle made simpler but not too simple, Physics Reports 1024 (2023) 1–29

    work page 2023

  45. [45]

    C. E. Rasmussen, C. K. Williams, Gaussian processes for machine learning, Vol. 2, MIT press Cam- bridge, MA, 2006

    work page 2006

  46. [46]

    C. E. Rasmussen, Z. Ghahramani, Bayesian Monte Carlo, Advances in neural information processing systems (2003) 505–512

    work page 2003

  47. [47]

    P.Wei, X.Zhang, M.Beer, Adaptiveexperimentdesignforprobabilisticintegration, ComputerMethods in Applied Mechanics and Engineering 365 (2020) 113035

    work page 2020

  48. [48]

    F.-X.Briol, C.J.Oates, M.Girolami, M.A.Osborne, D.Sejdinovic, Probabilisticintegration, Statistical Science 34 (1) (2019) 1–22

    work page 2019

  49. [49]

    F. Hong, P. Wei, M. Beer, Parallelization of adaptive Bayesian cubature using multimodal optimization algorithms, Engineering Computations 41 (2) (2024) 413–437

    work page 2024

  50. [50]

    Chevalier, D

    C. Chevalier, D. Ginsbourger, X. Emery, Corrected Kriging update formulae for batch-sequential data assimilation, in: Mathematics of Planet Earth: Proceedings of the 15th Annual Conference of the International Association for Mathematical Geosciences, Springer, 2013, pp. 119–122

    work page 2013

  51. [51]

    T. Zhou, T. Guo, Y. Dong, F. Yang, D. M. Frangopol, Look-ahead active learning reliability analysis based on stepwise margin reduction, Reliability Engineering & System Safety 243 (2024) 109830

    work page 2024

  52. [52]

    P.Wei, Y.Zheng, J.Fu, Y.Xu, W.Gao, Anexpectedintegratederrorreductionfunctionforaccelerating Bayesian active learning of failure probability, Reliability Engineering & System Safety 231 (2023) 108971. 46

    work page 2023

  53. [53]

    Wei, Transitional active learning of small probabilities, Computer Methods in Applied Mechanics and Engineering 444 (2025) 118144

    P. Wei, Transitional active learning of small probabilities, Computer Methods in Applied Mechanics and Engineering 444 (2025) 118144

    work page 2025

  54. [54]

    C. H. Bennett, Efficient estimation of free energy differences from Monte Carlo data, Journal of Com- putational Physics 22 (2) (1976) 245–268

    work page 1976

  55. [55]

    Katafygiotis, K

    L. Katafygiotis, K. Zuev, Estimation of small failure probabilities in high dimensions by adaptive linked importance sampling, COMPDYN 2007 (2007). 47

    work page 2007