Abstract
Approximate Bayesian computation (ABC) using a sequential Monte Carlo method provides a comprehensive platform for parameter estimation, model selection and sensitivity analysis in differential equations. However, this method, like other Monte Carlo methods, incurs a significant computational cost as it requires explicit numerical integration of differential equations to carry out inference. In this paper we propose a novel method for circumventing the requirement of explicit integration by using derivatives of Gaussian processes to smooth the observations from which parameters are estimated. We evaluate our methods using synthetic data generated from model biological systems described by ordinary and delay differential equations. Upon comparing the performance of our method to existing ABC techniques, we demonstrate that it produces comparably reliable parameter estimates at a significantly reduced execution time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beaumont, M.A., Cornuet, J.M., Marin, J.M., Robert, C.P.: Adaptive approximate Bayesian computation. Biometrika 96(4), 983–990 (2009)
Calderhead, B., Girolami, M., Lawrence, N.D.: Accelerating Bayesian inference over nonlinear differential equations with Gaussian processes. In: Proceedings of Advances in Neural Information Processing Systems 21, pp 217–224 (2008)
Del Moral, P., Doucet, A., Jasra, A.: Sequential Monte Carlo samplers. J. R. Stat. Soc. Ser. B 68(3), 411–436 (2006)
Del Moral, P., Doucet, A., Jasra, A.: An adaptive sequential Monte Carlo method for approximate Bayesian computation. Stat. Comput. 22(5), 1009–1020 (2012)
Didelot, X., Everitt, R.G., Johansen, A.M., Lawson, D.J.: Likelihood-free estimation of model evidence. Bayesian Anal. 6(1), 49–76 (2011)
Dondelinger, F., Filippone, M., Rogers, S., Husmeier, D.: ODE parameter inference using adaptive gradient matching with Gaussian processes. In: Proceedings of 16th International Conference on Artificial Intelligence and Statistics, pp 216–228 (2013)
Drovandi, C.C., Pettitt, A.N.: Estimation of parameters for macroparasite population evolution using approximate Bayesian computation. Biometrics 67(1), 225–233 (2011)
Filippi, S., Barnes, C.P., Cornebise, J., Stumpf, M.P.: On optimality of kernels for approximate Bayesian computation using sequential Monte Carlo. Stat. Appl. Genet. Mol. Biol. 12(1), 87–107 (2013)
MacKay, D.J.C.: Introduction to Gaussian processes. NATO ASI Ser. F 168, 133–166 (1998)
Monk, N.A.M.: Oscillatory expression of Hes1, p53, and NF-\(\kappa \)B driven by transcriptional time delays. Curr. Biol. 13, 1409–1413 (2003)
Murray, J.D.: Mathematical Biology: I. An Introduction. Springer, New York (2002)
Neal, R.M.: Regression and classification using Gaussian process priors. Bayesian Stat. 6, 475–501 (1998)
O’Hagan, A., Kingman, J.F.C.: Curve fitting and optimal design for prediction. J. R. Stat. Soc. Ser. B 40, 1–42 (1978)
Ramsay, J.O., Hooker, G., Campbell, D., Cao, J.: Parameter estimation for differential equations: a generalized smoothing approach. J. R. Stat. Soc. Ser. B 69(5), 741–796 (2007)
Rasmussen, C.E., Nickisch, H.: Gaussian processes for machine learning (GPML) toolbox. J. Mach. Learn. Res. 9999, 3011–3015 (2010)
Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)
Ruttor, A., Batz, P., Opper, M.: Approximate Gaussian process inference for the drift of stochastic differential equations. In: Proceedings of Advances in Neural Information Processing Systems 26, pp 2040–2048 (2013)
Sisson, S.A., Fan, Y., Tanaka, M.M.: Sequential Monte Carlo without likelihoods. Proc. Natl Acad. Sci. U.S.A. 104(6), 1760–1765 (2007)
Sisson, S.A., Fan, Y., Tanaka, M.M.: Correction for Sisson sequential Monte Carlo without likelihoods. Proc. Natl Acad. Sci. U.S.A. 106(39), 16,889 (2009)
Solak, E., Murray-Smith, R., Leithead, W., Rasmussen, C., Leith, D.: Derivative observations in Gaussian process models of dynamic systems. In: Proceedings of Advances in Neural Information Processing Systems 15, pp 1033–1040 (2002)
Toni, T., Welch, D., Strelkowa, N., Ipsen, A., Stumpf, M.P.: Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface 6(31), 187–202 (2009)
Varah, J.M.: A spline least squares method for numerical parameter estimation in differential equations. SIAM J. Sci. Stat. Comput. 3(1), 28–46 (1982)
Vyshemirsky, V., Girolami, M.A.: Bayesian ranking of biochemical system models. Bioinformatics 24(6), 833–839 (2008)
Wang, Y., Barber, D.: Gaussian processes for Bayesian estimation in ordinary differential equations. In: Proceedings of 31st International Conference on Machine Learning, pp 1485–1493 (2014)
Acknowledgments
We want to thank the anonymous reviewers for their valuable comments. This work was supported by project PLants Employed As SEnsor Devices (PLEASED), EC grant agreement number 296582.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ghosh, S., Dasmahapatra, S. & Maharatna, K. Fast approximate Bayesian computation for estimating parameters in differential equations. Stat Comput 27, 19–38 (2017). https://doi.org/10.1007/s11222-016-9643-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11222-016-9643-4