The Annals of Statistics

Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

Richard Nickl and Jakob Söhl

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider nonparametric Bayesian inference in a reflected diffusion model $dX_{t}=b(X_{t})\,dt+\sigma(X_{t})\,dW_{t}$, with discretely sampled observations $X_{0},X_{\Delta},\ldots,X_{n\Delta}$. We analyse the nonlinear inverse problem corresponding to the “low frequency sampling” regime where $\Delta>0$ is fixed and $n\to\infty$. A general theorem is proved that gives conditions for prior distributions $\Pi$ on the diffusion coefficient $\sigma$ and the drift function $b$ that ensure minimax optimal contraction rates of the posterior distribution over Hölder–Sobolev smoothness classes. These conditions are verified for natural examples of nonparametric random wavelet series priors. For the proofs, we derive new concentration inequalities for empirical processes arising from discretely observed diffusions that are of independent interest.

Article information

Source
Ann. Statist., Volume 45, Number 4 (2017), 1664-1693.

Dates
Received: October 2015
Revised: July 2016
First available in Project Euclid: 28 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1498636870

Digital Object Identifier
doi:10.1214/16-AOS1504

Mathematical Reviews number (MathSciNet)
MR3670192

Zentralblatt MATH identifier
06773287

Subjects
Primary: 62G05: Estimation
Secondary: 60J60: Diffusion processes [See also 58J65] 62F15: Bayesian inference 62G20: Asymptotic properties

Keywords
Nonlinear inverse problem Bayesian inference diffusion model

Citation

Nickl, Richard; Söhl, Jakob. Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions. Ann. Statist. 45 (2017), no. 4, 1664--1693. doi:10.1214/16-AOS1504. https://projecteuclid.org/euclid.aos/1498636870


Export citation

References

  • [1] Adamczak, R. (2008). A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13 1000–1034.
  • [2] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften 348. Springer, Cham.
  • [3] Baraud, Y. (2010). A Bernstein-type inequality for suprema of random processes with applications to model selection in non-Gaussian regression. Bernoulli 16 1064–1085.
  • [4] Bass, R. F. (1998). Diffusions and Elliptic Operators. Springer, New York.
  • [5] Bass, R. F. (2011). Stochastic Processes. Cambridge Series in Statistical and Probabilistic Mathematics 33. Cambridge Univ. Press, Cambridge.
  • [6] Baxendale, P. H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab. 15 700–738.
  • [7] Castillo, I. and Nickl, R. (2013). Nonparametric Bernstein–von Mises theorems in Gaussian white noise. Ann. Statist. 41 1999–2028.
  • [8] Castillo, I. and Nickl, R. (2014). On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures. Ann. Statist. 42 1941–1969.
  • [9] Ghosal, S., Ghosh, J. K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500–531.
  • [10] Ghosal, S. and van der Vaart, A. (2007). Convergence rates of posterior distributions for non-i.i.d. observations. Ann. Statist. 35 192–223.
  • [11] Giné, E. and Nickl, R. (2011). Rates on contraction for posterior distributions in $L^{r}$-metrics, $1\leq r\leq\infty$. Ann. Statist. 39 2883–2911.
  • [12] Giné, E. and Nickl, R. (2016). Mathematical Foundations of Infinite-Dimensional Statistical Models, Cambridge University Press, Cambridge.
  • [13] Gobet, E., Hoffmann, M. and Reiß, M. (2004). Nonparametric estimation of scalar diffusions based on low frequency data. Ann. Statist. 32 2223–2253.
  • [14] Golightly, A. and Wilkinson, D. J. (2005). Bayesian inference for stochastic kinetic models using a diffusion approximation. Biometrics 61 781–788.
  • [15] Gugushvili, S. and Spreij, P. (2014). Nonparametric Bayesian drift estimation for multidimensional stochastic differential equations. Lith. Math. J. 54 127–141.
  • [16] Koskela, J., Spano, D. and Jenkins, P. A. (2015). Consistency of Bayesian nonparametric inference for discretely observed jump diffusions. Preprint. Available at arXiv:1506.04709.
  • [17] Meyer, Y. (1992). Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37. Cambridge Univ. Press, Cambridge.
  • [18] Nickl, R. and Söhl, J. (2017). Supplement to “Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions.” DOI:10.1214/16-AOS1504SUPP.
  • [19] Papaspiliopoulos, O., Pokern, Y., Roberts, G. O. and Stuart, A. M. (2012). Nonparametric estimation of diffusions: A differential equations approach. Biometrika 99 511–531.
  • [20] Paulin, D. (2015). Concentration inequalities for Markov chains by Marton couplings and spectral methods. Electron. J. Probab. 20 1–32.
  • [21] Pokern, Y., Stuart, A. M. and van Zanten, J. H. (2013). Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs. Stochastic Process. Appl. 123 603–628.
  • [22] Ray, K. (2013). Bayesian inverse problems with non-conjugate priors. Electron. J. Stat. 7 2516–2549.
  • [23] Roberts, G. O. and Stramer, O. (2001). On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm. Biometrika 88 603–621.
  • [24] Söhl, J. and Trabs, M. (2016). Adaptive confidence bands for Markov chains and diffusions: Estimating the invariant measure and the drift. ESAIM Probab. Stat. 20 432–462.
  • [25] Stroock, D. W. and Varadhan, S. R. S. (1971). Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24 147–225.
  • [26] Stuart, A. M. (2010). Inverse problems: A Bayesian perspective. Acta Numer. 19 451–559.
  • [27] 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.
  • [28] Triebel, H. (2010). Theory of Function Spaces. Birkhäuser/Springer Basel AG, Basel.
  • [29] van Waaij, J. and van Zanten, H. (2016). Gaussian process methods for one-dimensional diffusions: Optimal rates and adaptation. Electron. J. Stat. 10 628–645.
  • [30] van Zanten, H. (2013). Nonparametric Bayesian methods for one-dimensional diffusion models. Math. Biosci. 243 215–222.
  • [31] van der Meulen, F., Schauer, M. and van Zanten, H. (2014). Reversible jump MCMC for nonparametric drift estimation for diffusion processes. Comput. Statist. Data Anal. 71 615–632.
  • [32] van der Meulen, F. and van Zanten, H. (2013). Consistent nonparametric Bayesian inference for discretely observed scalar diffusions. Bernoulli 19 44–63.
  • [33] van der Meulen, F. H., van der Vaart, A. W. and van Zanten, J. H. (2006). Convergence rates of posterior distributions for Brownian semimartingale models. Bernoulli 12 863–888.
  • [34] van der Vaart, A. W. and van Zanten, J. H. (2008). Rates of contraction of posterior distributions based on Gaussian process priors. Ann. Statist. 36 1435–1463.

Supplemental materials

  • The supplement to “Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions”. This supplement contains several proofs of results in the main paper and states and proves a proposition on Lipschitz properties of self-adjoint operators.