zbMATH — the first resource for mathematics

Iterative construction of Gaussian process surrogate models for Bayesian inference. (English) Zbl 1437.62105
Summary: A new algorithm is developed to tackle the issue of sampling non-Gaussian model parameter posterior probability distributions that arise from solutions to Bayesian inverse problems. The algorithm aims to mitigate some of the hurdles faced by traditional Markov Chain Monte Carlo (MCMC) samplers, through constructing proposal probability densities that are both, easy to sample and that provide a better approximation to the target density than a simple Gaussian proposal distribution would. To achieve that, a Gaussian proposal distribution is augmented with a Gaussian Process (GP) surface that helps capture non-linearities in the log-likelihood function. In order to train the GP surface, an iterative approach is adopted for the optimal selection of points in parameter space. Optimality is sought by maximizing the information gain of the GP surface using a minimum number of forward model simulation runs. The accuracy of the GP-augmented surface approximation is assessed in two ways. The first consists of comparing predictions obtained from the approximate surface with those obtained through running the actual simulation model at hold-out points in parameter space. The second consists of a measure based on the relative variance of sample weights obtained from sampling the approximate posterior probability distribution of the model parameters. The efficacy of this new algorithm is tested on inferring reaction rate parameters in a 3-node and 6-node network toy problems, which imitate idealized reaction networks in combustion applications.
62F15 Bayesian inference
60G15 Gaussian processes
68T05 Learning and adaptive systems in artificial intelligence
62P30 Applications of statistics in engineering and industry; control charts
EGO; emcee; Rtwalk; t-walk
Full Text: DOI
[1] Apte, Amit; Hairer, Martin; Stuart, A. M.; Voss, Jochen, Sampling the posterior: An approach to non-gaussian data assimilation, Physica D, 230, 1-2, 50-64 (2007) · Zbl 1113.62026
[2] Arridge, S. R.; Kaipio, J. P.; Kolehmainen, V.; Schweiger, M.; Somersalo, E.; Tarvainen, T.; Vauhkonen, M., Approximation errors and model reduction with an application in optical diffusion tomography, Inverse Problems, 22, 1, 175 (2006) · Zbl 1138.65042
[3] Berger, J. B.M. B.J.; Dawid, A.; Smith, D. H.A.; West, M., Markov chain monte carlo-based approaches for inference in computationally intensive inverse problems, (Bayesian Statistics 7: Proceedings of the Seventh Valencia International Meeting (2003), Oxford University Press), 181
[4] Bilionis, Ilias; Zabaras, Nicholas; Konomi, Bledar A.; Lin, Guang, Multi-output separable gaussian process: Towards an efficient, fully bayesian paradigm for uncertainty quantification, J. Comput. Phys., 241, 212-239 (2013) · Zbl 1349.76760
[5] Blight, B. J.N.; Ott, L., A bayesian approach to model inadequacy for polynomial regression, Biometrika, 62, 1, 79-88 (1975) · Zbl 0308.62062
[6] Bui-Thanh, Tan; Willcox, Karen; Ghattas, Omar, Model reduction for large-scale systems with high-dimensional parametric input space, SIAM J. Sci. Comput., 30, 6, 3270-3288 (2008) · Zbl 1196.37127
[7] Bui-Thanh, Tan; Willcox, Karen; Ghattas, Omar, Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications, AIAA J., 46, 10, 2520-2529 (2008)
[8] Busby, Daniel, Hierarchical adaptive experimental design for gaussian process emulators, Reliab. Eng. Syst. Saf., 94, 7, 1183-1193 (2009), Special Issue on Sensitivity Analysis
[9] Cai, Bo; Meyer, Renate; Perron, François, Metropolis-hastings algorithms with adaptive proposals, Stat. Comput., 18, 4, 421-433 (2008)
[10] Christen, J. Andrés; Fox, Colin, Markov chain monte carlo using an approximation, J. Comput. Graph. Stat., 14, 4, 795-810 (2005)
[11] Christen, J. Andrés; Fox, Colin, A general purpose sampling algorithm for continuous distributions (the t-walk), Bayesian Anal., 5, 2, 263-281 (2010) · Zbl 1330.62007
[12] Christen, J. Andrs; Sans, Bruno, Advances in the sequential design of computer experiments based on active learning, Comm. Statist. Theory Methods, 40, 24, 4467-4483 (2011) · Zbl 1318.62264
[13] Cohn, David A., Neural network exploration using optimal experiment design, Neural Netw., 9, 6, 1071-1083 (1996)
[14] Cohn, David A., Minimizing statistical bias with queries, (Advances in Neural Information Processing Systems (1997)), 417-423
[15] Cohn, David A.; Ghahramani, Zoubin; Jordan, Michael I., Active learning with statistical models, J. Artif. Intell. Res., 4, 129-145 (1996) · Zbl 0900.68366
[16] Conrad, Patrick R.; Marzouk, Youssef M.; Pillai, Natesh S.; Smith, Aaron, Accelerating asymptotically exact mcmc for computationally intensive models via local approximations, J. Amer. Statist. Assoc., 111, 516, 1591-1607 (2016)
[17] Cui, Tiangang; Law, Kody J. H.; Marzouk, Youssef M., Dimension-independent likelihood-informed mcmc, J. Comput. Phys., 304, 109-137 (2016) · Zbl 1349.65009
[18] Currin, Carla, A bayesian approach to the design and analysis of computer experiments (1988), ORNL Oak Ridge National Laboratory (US)
[19] Currin, Carla; Mitchell, Toby; Morris, Max; Ylvisaker, Don, Bayesian prediction of deterministic functions, with applications to the design and analysis of computer experiments, J. Amer. Statist. Assoc., 86, 416, 953-963 (1991)
[20] Dostert, Paul; Efendiev, Yalchin; Hou, Thomas Y.; Luo, Wuan, Coarse-gradient langevin algorithms for dynamic data integration and uncertainty quantification, J. Comput. Phys., 217, 1, 123-142 (2006) · Zbl 1146.76637
[21] Efendiev, Yalchin; Hou, Thomas; Luo, Wuan, Preconditioning markov chain monte carlo simulations using coarse-scale models, SIAM J. Sci. Comput., 28, 2, 776-803 (2006) · Zbl 1111.65003
[22] El Moselhy, Tarek A.; Marzouk, Youssef M., Bayesian inference with optimal maps, J. Comput. Phys., 231, 23, 7815-7850 (2012) · Zbl 1318.62087
[23] Fedorov, Valerii Vadimovich, Theory of Optimal Experiments (1972), Elsevier
[24] Foreman-Mackey, Daniel; Hogg, David W.; Lang, Dustin; Goodman, Jonathan, Emcee: the mcmc hammer, Publ. Astron. Soc. Pac., 125, 925, 306 (2013)
[25] Geweke, John; Tanizaki, Hisashi, On markov chain monte carlo methods for nonlinear and non-gaussian state-space models, Comm. Statist. Simulation Comput., 28, 4, 867-894 (1999) · Zbl 0968.62541
[26] Ghanem, Roger G.; Spanos, Pol D., Stochastic finite element method: Response statistics, (Stochastic Finite Elements: A Spectral Approach (1991), Springer), 101-119
[27] Gilks, W. R.; Best, N. G.; Tan, K. K.C., Adaptive rejection metropolis sampling within gibbs sampling, J. R. Stat. Soc. Ser. C. Appl. Stat., 44, 4, 455-472 (1995) · Zbl 0893.62110
[28] Girolami, Mark; Calderhead, Ben, Riemann manifold langevin and hamiltonian monte carlo methods, J. R. Stat. Soc. Ser. B Stat. Methodol., 73, 2, 123-214 (2011) · Zbl 1411.62071
[29] Goodman, Jonathan; Weare, Jonathan, Ensemble samplers with affine invariance, Commun. Appl. Math. Comput. Sci., 5, 1, 65-80 (2010) · Zbl 1189.65014
[30] Gramacy, Robert B.; Apley, Daniel W., Local gaussian process approximation for large computer experiments, J. Comput. Graph. Statist., 24, 2, 561-578 (2015)
[31] Gramacy, Robert B.; Lee, Herbert K. H., Bayesian treed gaussian process models with an application to computer modeling, J. Amer. Statist. Assoc., 103, 483, 1119-1130 (2008) · Zbl 1205.62218
[32] Gramacy, Robert B.; Lee, Herbert K. H., Adaptive design and analysis of supercomputer experiments, Technometrics, 51, 2, 130-145 (2009)
[33] Gramacy, Robert B.; Lee, Herbert K. H.; Macready, William G., Parameter space exploration with gaussian process trees, (Proceedings of the Twenty-First International Conference on Machine Learning (2004), ACM), 45
[34] Haario, Heikki; Laine, Marko; Mira, Antonietta; Saksman, Eero, Dram: efficient adaptive mcmc, Stat. Comput., 16, 4, 339-354 (2006)
[35] Habib, Salman; Heitmann, Katrin; Higdon, David; Nakhleh, Charles; Williams, Brian, Cosmic calibration: Constraints from the matter power spectrum and the cosmic microwave background, Phys. Rev. D, 76, 8, 083503 (2007)
[36] Heitmann, Katrin; White, Martin; Wagner, Christian; Habib, Salman; Higdon, David, The coyote universe. i. precision determination of the nonlinear matter power spectrum, Astrophys. J., 715, 1, 104 (2010)
[37] Higdon, Dave; Gattiker, James; Williams, Brian; Rightley, Maria, Computer model calibration using high-dimensional output, J. Amer. Statist. Assoc., 103, 482, 570-583 (2008) · Zbl 05564511
[38] Iglesias, Marco; Stuart, Andrew M., Inverse problems and uncertainty quantification, SIAM News, July/August (2014)
[39] Jones, Donald R.; Schonlau, Matthias; Welch, William J., Efficient global optimization of expensive black-box functions, J. Glob. Optim., 13, 4, 455-492 (1998) · Zbl 0917.90270
[40] Kandasamy, Kirthevasan; Schneider, Jeff; Póczos, Barnabás, Query efficient posterior estimation in scientific experiments via bayesian active learning, Artificial Intelligence, 243, 45-56 (2017) · Zbl 1402.62044
[41] Kennedy, Marc C.; Anderson, Clive W.; Conti, Stefano; O’Hagan, Anthony, Case studies in gaussian process modelling of computer codes, Reliab. Eng. Syst. Saf., 91, 10-11, 1301-1309 (2006)
[42] Kennedy, Marc C.; O’Hagan, Anthony, Bayesian calibration of computer models, J. R. Stat. Soc. Ser. B Stat. Methodol., 63, 3, 425-464 (2001) · Zbl 1007.62021
[43] Koehler, J. R.; Owen, A. B., 9 computer experiments, Handb. Statist., 13, 261-308 (1996) · Zbl 0919.62089
[44] Le Maître, Olivier; Knio, Omar M., Spectral Methods for Uncertainty Quantification: with Applications to Computational Fluid Dynamics (2010), Springer Science & Business Media · Zbl 1193.76003
[45] MacKay, David J. C., Information-based objective functions for active data selection, Neural Comput., 4, 4, 590-604 (1992)
[46] MacKay, David J. C., Introduction to gaussian processes, NATO ASI Series F Comput. Syst. Sci., 168, 133-166 (1998) · Zbl 0936.68081
[47] Martin, James; Wilcox, Lucas C.; Burstedde, Carsten; Ghattas, Omar, A stochastic newton mcmc method for large-scale statistical inverse problems with application to seismic inversion, SIAM J. Sci. Comput., 34, 3, A1460-A1487 (2012) · Zbl 1250.65011
[48] Martino, Luca; Casarin, Roberto; Leisen, Fabrizio; Luengo, David, Adaptive independent sticky mcmc algorithms, EURASIP J. Adv. Signal Process., 2018, 1, 5 (2018)
[49] Martino, L.; Read, J.; Luengo, D., Independent doubly adaptive rejection metropolis sampling within gibbs sampling, IEEE Trans. Signal Process., 63, 12, 3123-3138 (2015) · Zbl 1394.94828
[50] Martino, L.; Vicent, J.; Camps-Valls, G., Automatic emulator and optimized look-up table generation for radiative transfer models, (2017 IEEE International Geoscience and Remote Sensing Symposium (IGARSS) (2017)), 1457-1460
[51] Marzouk, Youssef M.; Najm, Habib N.; Rahn, Larry A., Stochastic spectral methods for efficient bayesian solution of inverse problems, J. Comput. Phys., 224, 2, 560-586 (2007) · Zbl 1120.65306
[52] Meyer, Renate; Cai, Bo; Perron, Franois, Adaptive rejection metropolis sampling using lagrange interpolation polynomials of degree 2, Comput. Statist. Data Anal., 52, 7, 3408-3423 (2008) · Zbl 1452.62099
[53] Morris, Max D.; Mitchell, Toby J.; Ylvisaker, Donald, Bayesian design and analysis of computer experiments: use of derivatives in surface prediction, Technometrics, 35, 3, 243-255 (1993) · Zbl 0785.62025
[54] Morzfeld, Matthias; Day, Marcus S.; Grout, Ray W.; Pau, George Shu Heng; Finsterle, Stefan A.; Bell, John B., Iterative importance sampling algorithms for parameter estimation, SIAM J. Sci. Comput. (2018) · Zbl 1385.65007
[55] Narayanan, Velamur Asokan Badri; Zabaras, Nicholas, Stochastic inverse heat conduction using a spectral approach, Internat. J. Numer. Methods Engrg., 60, 9, 1569-1593 (2004) · Zbl 1098.80008
[56] Neal, Radford M., Mcmc using hamiltonian dynamics, Handb. Markov Chain Monte Carlo, 2, 11 (2011) · Zbl 1229.65018
[57] Oakley, Jeremy; O’hagan, Anthony, Bayesian inference for the uncertainty distribution of computer model outputs, Biometrika, 89, 4, 769-784 (2002)
[58] OHagan, A., Bayesian analysis of computer code outputs: A tutorial, Reliab. Eng. Syst. Saf., 91, 10, 1290-1300 (2006), The Fourth International Conference on Sensitivity Analysis of Model Output (SAMO 2004)
[59] O’Hagan, Anthony; Kingman, John Frank Charles, Curve fitting and optimal design for prediction, J. R. Stat. Soc. Ser. B Stat. Methodol., 1-42 (1978) · Zbl 0374.62070
[60] Paass, Gerhard; Kindermann, Jörg, Bayesian query construction for neural network models, (Advances in Neural Information Processing Systems (1995)), 443-450
[61] Parno, Matthew D.; Marzouk, Youssef M., Transport map accelerated markov chain monte carlo, SIAM/ASA J. Uncertain. Quantif., 6, 2, 645-682 (2018) · Zbl 1394.65004
[62] Preuss, Roland; von Toussaint, Udo, Global optimization employing gaussian process-based bayesian surrogates, Entropy, 20, 3, 201 (2018)
[63] Rasmussen, Carl E., Gaussian processes to speed up hybrid monte carlo for expensive bayesian integrals (2003)
[64] Rasmussen, Carl Edward; Williams, Christopher K. I., Gaussian Process for Machine Learning (2006), MIT press · Zbl 1177.68165
[65] Roberts, Gareth O.; Tweedie, Richard L., Exponential convergence of langevin distributions and their discrete approximations, Bernoulli, 2, 4, 341-363 (1996) · Zbl 0870.60027
[66] Rozza, Gianluigi; Huynh, Dinh Bao Phuong; Patera, Anthony T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods Eng., 15, 3, 1 (2007)
[67] Sacks, Jerome; Welch, William J.; Mitchell, Toby J.; Wynn, Henry P., Design and analysis of computer experiments, Stat. Sci., 409-423 (1989) · Zbl 0955.62619
[68] Seeger, Matthias; Williams, Christopher; Lawrence, Neil, Fast forward selection to speed up sparse Gaussian process regression, Artif. Intell. Stat., 9 (2003), number EPFL-CONF-161318
[69] Seo, Sambu; Wallat, Marko; Graepel, Thore; Obermayer, Klaus, Gaussian process regression: Active data selection and test point rejection, (Mustererkennung 2000 (2000), Springer), 27-34
[70] Shao, Wei; Guo, Guangbao; Meng, Fanyu; Jia, Shuqin, An efficient proposal distribution for metropolishastings using a b-splines technique, Comput. Statist. Data Anal., 57, 1, 465-478 (2013) · Zbl 1365.65026
[71] Tarantola, Albert, Inverse Problem Theory and Methods for Model Parameter Estimation, Vol. 89 (2005), siam · Zbl 1074.65013
[72] Tarantola, Albert, Popper, bayes and the inverse problem, Nature physics, 2, 8, 492 (2006)
[73] Vanden-Eijnden, Eric; Weare, Jonathan, Data assimilation in the low noise regime with application to the kuroshio, Mon. Weather Rev., 141, 6, 1822-1841 (2013)
[74] Wang, Zheng; Yan, Shuicheng; Zhang, Changshui, Active learning with adaptive regularization, Pattern Recognit., 44, 10, 2375-2383 (2011), Semi-Supervised Learning for Visual Content Analysis and Understanding · Zbl 1218.68137
[75] Wang, Jingbo; Zabaras, Nicholas, Using bayesian statistics in the estimation of heat source in radiation, Int. J. Heat Mass Transfer, 48, 1, 15-29 (2005) · Zbl 1122.80307
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.