Stochastic convergence rates and applications of adaptive quadrature in Bayesian inference. (English) Zbl 07820413
Summary: We provide the first stochastic convergence rates for a family of adaptive quadrature rules used to normalize the posterior distribution in Bayesian models. Our results apply to the uniform relative error in the approximate posterior density, the coverage probabilities of approximate credible sets, and approximate moments and quantiles, therefore, guaranteeing fast asymptotic convergence of approximate summary statistics used in practice. The family of quadrature rules includes adaptive Gauss-Hermite quadrature, and we apply this rule in two challenging low-dimensional examples. Further, we demonstrate how adaptive quadrature can be used as a crucial component of a modern approximate Bayesian inference procedure for high-dimensional additive models. The method is implemented and made publicly available in the aghq package for the R language, available on CRAN. Supplementary materials for this article are available online.
MSC:
| 62-XX | Statistics |
Keywords:
adaptive quadrature; approximate inference; asymptotic convergence; Bayesian; higher-order; spatialReferences:
| [1] | Almutiry, W., Warriyar, K. V., and Deardon, R. (2020), “Continuous Time Individual-Level Models of Infectious Disease: EpiILMCT,” arXiv:2006.00135v1. |
| [2] | Bojanov, B., and Petrov, P. P. (2001), “Gaussian Interval Quadrature Formula,” Numerische Mathematik, 87, 625-643. · Zbl 0969.41018 |
| [3] | Brown, P. (2011), “Model-based Geostatistics the Easy Way,” Journal of Statistical Software, 73, 423-498. |
| [4] | Cagnone, S., and Monari, P. (2013), “Latent Variable Models for Ordinal Data by using the Adaptive Quadrature Approximation,” Computational Statistics, 28, 597-619. DOI: . · Zbl 1305.65028 |
| [5] | Davis, P. J., and Rabinowitz, P. (1975), Methods of Numerical Integration, New York: Academic Press. · Zbl 0304.65016 |
| [6] | Dick, J., Gantner, R. N., Le Gia, Q. T., and Schwab, C. (2019), “Higher Order Quasi-Monte Carlo Integration for Bayesian PDE Inversion,” Computers and Mathematics with Applications, 77, 144-172. DOI: . · Zbl 1442.62051 |
| [7] | Diggle, P. J., and Giorgi, E. (2016), “Model-based Geostatistics for Prevalence Mapping in Low-Resource Settings,” Journal of the American Statistical Association, 111, 1096-1120. DOI: . |
| [8] | Eadie, G. M., and Harris, W. E. (2016), “Bayesian Mass Estimates of the Milky Way: The Dark and Light Sides of Parameter Assumptions,” The Astrophysical Journal, 829. DOI: . |
| [9] | Fuglstad, G.-A., Simpson, D., Lindgren, F., and Rue, H. (2019), “Constructing Priors that Penalize the Complexity of Gaussian Random Fields,” Journal of the American Statistical Association, 114, 445-452. DOI: . · Zbl 1478.62279 |
| [10] | Geirsson, O. P., Hrafnkelsson, B., Simpson, D., and Sigurdarson, H. (2020), “LGM Split Sampler: An Efficient MCMC Sampling Scheme for Latent Gaussian Models,” Statistical Science, 35, 218-233. DOI: . · Zbl 07292510 |
| [11] | Genz, A., and Keister, B. D. (1996), “Fully Symmetric Interpolatory Rules for Multiple Integrals over Infinite Regions with Gaussian Weight,” Journal of Computational and Applied Mathematics, 71, 299-309. DOI: . · Zbl 0856.65011 |
| [12] | Giorgi, E., and Diggle, P. J. (2017), “PrevMap: An R Package for Prevalence Mapping,” Journal of Statistical Software, 78, 1-29. DOI: . |
| [13] | Giorgi, E., Schluter, D. K., and Diggle, P. J. (2018), “Bivariate Geostatistical Modelling of the Relationship between Loa loa Prevalence and Intensity of Infection,” Environmetrics, 29, e2447. DOI: . |
| [14] | Heiss, F., and Winschel, V. (2008), “Likelihood Approximation by Numerical Integration on Sparse Grids,” Journal of Econometrics, 144, 62-80. DOI: . · Zbl 1418.62466 |
| [15] | Hoffman, M. D., and Gelman, A. (2014), “The no-U-turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo,” Journal of Machine Learning Research, 15, 1593-1623. · Zbl 1319.60150 |
| [16] | Jin, S., and Andersson, B. (2020), “A Note on the Accuracy of Adaptive Gauss-Hermite Quadrature,” Biometrika, 107, 737-744. DOI: . · Zbl 1451.62092 |
| [17] | Kass, R., and Steffey, D. (1989), “Approximate Bayesian Inference in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models),” Journal of the American Statistical Association, 84, 717-726. DOI: . |
| [18] | Kass, R. E., Tierney, L., and Kadane, J. B. (1990), “The Validity of Posterior Expansions based on Laplace’s Method,” in Bayesian and Likelihood Methods in Statistics and Econometrics, eds. Geisser, S., Hodges, J.S., Press, S.J., and Zellner, A., pp. 473-488, Amsterdam: North Holland. · Zbl 0734.62034 |
| [19] | Kristensen, K., Nielson, A., Berg, C. W., Skaug, H., and Bell, B. M. (2016), “TMB: Automatic Differentiation and Laplace Approximation,” Journal of Statistical Software, 70, 1-21. DOI: . |
| [20] | Liu, Q., and Pierce, D. A. (1994), “A Note on Gauss-Hermite Quadrature,” Biometrika, 81, 624-629. · Zbl 0813.65053 |
| [21] | Monnahan, C., and Kristensen, K. (2018), “No-U-turn Sampling for Fast Bayesian Inference in ADMB and TMB: Introducing the adnuts and tmbstan R Packages,” PloS One, 13, e0197954. DOI: . |
| [22] | Naylor, J., and Smith, A. F. M. (1982), “Applications of a Method for the Efficient Computation of Posterior Distributions,”Journal of the Royal Statistical Society, Series C, 31, 214-225. DOI: . · Zbl 0521.65017 |
| [23] | Pinheiro, J. C., and Bates, D. M. (1995), “Approximations to the Log-Likelihood Function in the Nonlinear Mixed-Effects Model,” Journal of computational and Graphical Statistics, 4, 12-35. |
| [24] | Rue, H. (2001), “Fast Sampling of Gaussian Markov Random Fields,” Journal of the Royal Statistical Society, Series B, 63, 325-338. DOI: . · Zbl 0979.62075 |
| [25] | Rue, H., Martino, S., and Chopin, N. (2009), “Approximate Bayesian Inference for Latent Gaussian Models by using Integrated Nested Laplace Approximations,” Journal of the Royal Statistical Society, Series B, 71, 319-392. DOI: . · Zbl 1248.62156 |
| [26] | Schillings, C., and Schwab, C. (2016), “Scaling Limits in Computational Bayesian Inversion,” ESAIM: Mathematical Modelling and Numerical Analysis, 50, 1825-1856. DOI: . · Zbl 1358.65013 |
| [27] | Schlather, M., Malinowski, A., Menck, P. J., Oesting, M., and Strokorb, K. (2015), “Analysis, Simulation and Prediction of Multivariate Random Fields with Package Randomfields,” Journal of Statistical Software, 63, 1-25. DOI: . |
| [28] | Stringer, A., Brown, P., and Stafford, J. (2021), “Approximate Bayesian Inference for Case Crossover Models,” Biometrics, 77, 785-795. DOI: . · Zbl 1520.62332 |
| [29] | Taylor, B. M., and Diggle, P. J. (2014), “INLA or MCMC? A Tutorial and Comparative Evaluation for Spatial Prediction in Log-Gaussian Cox Processes,” Journal of Statistical Computation and Simulation, 84, 2266-2284. DOI: . · Zbl 1453.62214 |
| [30] | Tierney, L., and Kadane, J. B. (1986), “Accurate Approximations for Posterior Moments and Marginal Densities,” Journal of the American Statistical Association, 81, 82-86. DOI: . · Zbl 0587.62067 |
| [31] | van der Vaart, A. W. (1998), Asymptotic Statistics, Cambridge: Cambridge University Press. · Zbl 0910.62001 |
| [32] | Wachter, A., and Biegler, L. T. (2006), “On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming,” Mathematical Programming, 106, 25-57. DOI: . · Zbl 1134.90542 |
| [33] | Wood, S. (2020), “Simplified Integrated Nested Laplace Approximation,” Biometrika, 107, 223-230. · Zbl 1435.62341 |
| [34] | Wood, S., Pya, N., and Säfken, B. (2016), “Smoothing Parameter and Model Selection for General Smooth Models,” Journal of the American Statistical Association, 111, 1548-1575. DOI: . |
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