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Rejoinder: “Probabilistic integration: a role in statistical computation?”. (English) Zbl 1420.62136
Summary: This article is the rejoinder to the comments [F. J. Hickernell and R. Jagadeeswaran, ibid. 34, No. 1, 23–28 (2019; Zbl 1420.62139); A. B. Owen, ibid. 34, No. 1, 29–33 (2019; Zbl 1420.62145); M. L. Stein and Y. Hung, ibid. 34, No. 1, 34–37 (2019; Zbl 1420.62150)] on the authors’ paper [ibid. 34, No. 1, 1–22 (2019; Zbl 1420.62135)]. We would first like to thank the reviewers and many of our colleagues who helped shape this paper, the Editor for selecting our paper for discussion, and of course all of the discussants for their thoughtful, insightful and constructive comments. In this rejoinder, we respond to some of the points raised by the discussants and comment further on the fundamental questions underlying the paper: (i) Should Bayesian ideas be used in numerical analysis? and (ii) If so, what role should such approaches have in statistical computation?
MSC:
62G05 Nonparametric estimation
65D30 Numerical integration
65C60 Computational problems in statistics (MSC2010)
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[1] Briol, F-X., Oates, C. J., Girolami, M., Osborne, M. A. and Sejdinovic, D. (2019). Probabilistic integration: A role in statistical computation? Statist. Sci.34 1-22.
[2] Cockayne, J., Oates, C. J., Sullivan, T. and Girolami, M. (2017). Bayesian probabilistic numerical methods. arXiv:1701.04006.
[3] Cotter, S. L., Roberts, G. O., Stuart, A. M. and White, D. (2013). MCMC methods for functions: Modifying old algorithms to make them faster. Statist. Sci.28 424-446. · Zbl 1331.62132
[4] Cox, D. D. (1993). An analysis of Bayesian inference for nonparametric regression. Ann. Statist.21 903-923. · Zbl 0778.62003
[5] Diaconis, P. (1988). Bayesian numerical analysis. In Statistical Decision Theory and Related Topics, IV, Vol. 1 (West Lafayette, Ind., 1986) 163-175. Springer, New York.
[6] Diaconis, P. and Freedman, D. (1986). On the consistency of Bayes estimates. Ann. Statist.14 1-67. · Zbl 0595.62022
[7] Dunlop, M. M., Girolami, M., Stuart, A. M. and Teckentrup, A. L. (2018). How deep are deep Gaussian processes? J. Mach. Learn. Res.19 1-46. · Zbl 06982345
[8] Jagadeeswaran, R. and Hickernell, F. J. (2018). Fast automatic Bayesian cubature using lattice sampling. arXiv:1809.09803.
[9] Kanagawa, M., Sriperumbudur, B. and Fukumizu, K. (2016). Convergence guarantees for kernel-based quadrature rules in misspecified settings. In Advances in Neural Information Processing Systems 3288-3296.
[10] Kanagawa, M., Hennig, P., Sejdinovic, D. and Sriperumbudur, B. K. (2018). Gaussian processes and kernel methods: A review on connections and equivalences. arXiv:1807.02582.
[11] Karvonen, T. and Särkkä, S. (2018). Fully symmetric kernel quadrature. SIAM J. Sci. Comput.40 A697-A720.
[12] Oates, C. J., Cockayne, J. and Aykroyd, R. G. (2017). Bayesian probabilistic numerical methods for industrial process monitoring. arXiv:1707.06107.
[13] Owhadi, H., Scovel, C. and Sullivan, T. (2015). On the brittleness of Bayesian inference. SIAM Rev.57 566-582. · Zbl 1341.62094
[14] Ritter, K. (2000). Average-Case Analysis of Numerical Problems. Lecture Notes in Math.1733. Springer, Berlin. · Zbl 0949.65146
[15] Sniekers, S. and van der Vaart, A. (2015). Adaptive Bayesian credible sets in regression with a Gaussian process prior. Electron. J. Stat.9 2475-2527. · Zbl 1327.62300
[16] Szabó, B., van der Vaart, A. W. and van Zanten, J. H. (2015). Frequentist coverage of adaptive nonparametric Bayesian credible sets. Ann. Statist.43 1391-1428. · Zbl 1317.62040
[17] Xi, X., Briol, F.-X. and Girolami, M. (2018). Bayesian quadrature for multiple related integrals. In International Conference on Machine Learning, PMLR 80 5369-5378.
[18] Zhu, Z. and Stein, M. L. (2006). Spatial sampling design for prediction with estimated parameters. J. Agric. Biol. Environ. Stat.11 24-44.
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