Bayesian Analysis

Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami

Martin Lysy

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The authors present an ingenious probabilistic numerical solver for deterministic differential equations (DEs). The true solution is progressively identified via model interrogations, in a formal framework of Bayesian updating. I have attempted to extend the authors’ ideas to stochastic differential equations (SDEs), and discuss two challenges encountered in this endeavor: (i) the non-differentiability of SDE sample paths, and (ii) the sampling of diffusion bridges, typically required of solutions to the SDE inverse problem.

Article information

Bayesian Anal., Volume 11, Number 4 (2016), 1269-1273.

First available in Project Euclid: 30 November 2016

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stochastic differential equations probabilistic solution diffusion bridge sampling


Lysy, Martin. Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami. Bayesian Anal. 11 (2016), no. 4, 1269--1273. doi:10.1214/16-BA1036.

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  • Andrieu, C., Doucet, A., and Holenstein, R. (2010). “Particle Markov chain Monte Carlo Methods.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72: 1–33.
  • Bladt, M., Finch, S., and Sørensen, M. (2016). “Simulation of multivariate diffusion bridges.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(2): 343–369.
  • Chkrebtii, O. A., Campbell, D. A., Calderhead, B., and Girolami, M. A. (2016). “Bayesian solution uncertainty quantification for differential equations.” Bayesian Analysis.
  • Lysy, M. and Pillai, N. S. (2013). “Statistical Inference for Stochastic Differential Equations with Memory.” Technical report, University of Waterloo.
  • Roberts, G. O. and Stramer, O. (2001). “On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm.” Biometrika, 88(3): 603–621.
  • Van Kampen, N. G. (1981). “Itô versus Stratonovich.” Journal of Statistical Physics, 24(1): 175–187.

See also

  • Related item: Oksana A. Chkrebtii, David A. Campbell, Ben Calderhead, Mark A. Girolami (2016). Bayesian Solution Uncertainty Quantification for Differential Equations. Bayesian Anal. Vol. 11, Iss. 4, 1239–1267.