# A06 – Approximative Bayesian inference and model selection for stochastic differential equations (SDEs)

## Objectives

This project is concerned with Bayesian semi-parametric and fully nonparametric approaches for estimating drift functions in systems of stochastic differential equations (SDEs). We will develop robust and computational efficient sequential Monte Carlo approaches and variational Bayesian methods and will study their convergence rates and approximation properties. We will also derive new methods for Bayesian model selection to decide on the base of available data which prior from a given collection is most appropriate for the SDE estimation problem at hand.

The project will be carried out jointly between the

- TU Berlin, where the focus will be on variational Bayesian methods on combined state and drift estimation for SDEs,
- the Weierstrass-Institut Berlin, where the focus will be on prior selection for semi- and non-parametric statistics applied to SDEs, and
- the University of Potsdam, where the focus will be on sequential Monte Carlo methods for high-dimensional inference problems arising from SDEs.

The project will closely collaborate with project A01 (statistics of SPDEs), project A02 (stability and accuracy of particle filters) and several of the more applied projects from Research Area B.

### References

- Probabilistic Forecasting and Bayesian Data Assimilation
Colin Cotter and Sebastian Reich, Cambridge University Press, 2015

- Variational mean-field algorithm for efficient inference in large systems of SDEs
Michail Vrettas, Manfred Opper and Dan Cornford, Phys. Ref. E. 91:012145, 2015

- Bootstrap tuning of ordered model selection
Vladimir Spokoiny and Niklas Willrich, arXiv:1507.05034, Weierstrass-Institute Berlin, 2015

## Preprints

Reich, S., and Rozdeba, P. J. (2020).

*Posterior contraction rates for non-parametric state and drift estimation.*arXiv:2003.09219Avanesov, V. (2019).

*How to gamble with non-stationary X-armed bandits and have no regrets*. arXiv:1908.07636Avanesov, V. (2019).

*Structural break analysis in high-dimensional covariance structure.*arXiv: 1803.00508Avanesov, V. (2019).

*Nonparametric Change Point Detection in Regression*. arXiv:1903.02603Ty, A.J.A., Fang, Z., Gonzales, R.A., Rozdeba, P.J. and Abarbanel, H.D.I. (2019).

*Machine Learning of Time Series Using Time-delay Embedding and Precision Annealing.*arXiv:1902.05062Pathiraja, S. and van Leeuwen, P.J. (2018).

*Model uncertainty estimation in data assimilation for multi-scale systems with partially observed resolved variables*, Quarterly Journal of the Royal Meteorological Society, under review, arXiv: 1807.09621

## Publications

Garbuno-Inigo, A., Nüsken, N., and Reich, S. (2019).

*Affine invariant interacting Langevin dynamics for Bayesian inference.*To appear in SIAM J. Dyn. Syst. arXiv:1912.02859Mariucci, E., Ray, K. and Szabó, B. (2019).

*A Bayesian nonparametric approach to log-concave density estimation.*To appear in Bernoulli. arXiv: 1703.09531Götze, F., Naumov, A., Spokoiny, V. and Ulyanov, V. (2019).

*Gaussian comparison and anti-concentration inequalities for norms of Gaussian random elements*, Bernoulli, in print. arXiv:1708.08663Nüsken, N., Reich, S. and Rozdeba, P. J. (2019).

*State and parameter estimation from observed signal increments,*Entropy, Vol. 21(5), 505. arXiv:1903.10717 ; doi: 10.3390/e21050505Donner, C. and Opper, M. (2018).

*Efficient Bayesian Inference of Sigmoidal Gaussian Cox Processes,*Journal of Machine Learning Research 19, no 67, 1-34. Open AccessDonner, C. and Opper, M. (2018).

*Efficient Bayesian Inference for a Gaussian Process Density Model,*Proc. in Conference on Uncertainty in Artificial Intelligence, 2018. Open AccessSilin, I. and Spokoiny, V. (2018).

*Bayesian inference for spectral projectors of the covariance matrix,*Electron. J. Statist. 12(1), 1948-1987. doi:10.1214/18-EJS1451; arXiv:1711.11532