## Minicourse at Engineering Department of Oxford University in 13.11. - 2.12.2013

# Stochastic Differential Equations in Bayesian Dynamic Models and Machine Learning

**Teacher:**
Dr. Simo Särkkä,
Aalto University, Finland. Visiting scholar at the Dept. of Statistics
of Oxford University.

**Coordinator:**
Dr. Michael A. Osborne,
University of Oxford.

**Schedule:** Lectures in LR7/LR8 on Wednesdays and
Thursdays 3-5pm (13.11., 14.11., 20.11., 21.11., 27.11., 28.11.).
Exercises in LR7 on Mondays 11am-1pm (18.11., 25.11., 28.11.,
2.12.). Bring a laptop with Matlab/Octave to the exercise
session.

Location: Oxford University, Department of Engineering Science, LR7/LR8.

**Topic:** An introduction to the theory, applications
and numerical methods for SDEs. Application to Bayesian estimation of
in continuous-time models, Gaussian processes in machine learning, and
to modeling of physical systems. After the course the student should
be able to formulate a simple SDE model for an application, analyze
its properties, and solve it numerically using appropriate
methods. The student should also be familiar with the basic principles
of Bayesian estimation in SDE models and their use in Gaussian process
regression.

**Target audience:** Advanced undergraduate and graduate
(PhD) students. Researchers and engineers wishing to get a hands-on
introduction to the topic.

**Prerequisites:**
Multivariate differential and integral
calculus, matrix analysis, basic probability, Matlab/Octave.

## Lectures:

The slides are copied here after each lecture (the topics are preliminary and might change):

**LR7 - Wednesday, 13th November from 3-5pm:***Pragmatic Introduction to Stochastic Differential Equations*(Slides as PDF)**LR7 - Thursday, 14th November from 3-5pm:***Ito Calculus and Stochastic Differential Equations*(Slides as PDF)**LR7 - Wednesday, 20th November from 3-5pm:***Probability Distributions and Statistics of SDEs*(Slides as PDF)**LR8 - Thursday, 21st November from 3-5pm:***Numerical solutions of SDEs*(Slides as PDF)**LR7 - Wednesday, 27th November from 3-5pm:***Bayesian Inference in SDE Models*(Slides as PDF)**LR7 - Thursday, 28th November from 3-5pm:***State-Space Inference in Gaussian Process Regression*(Slides as PDF)

## Exercise hours:

The exercises are interactive demonstration sessions where mainly the lecturer is solving the problems on white board and/or in Matlab. Sometimes volunteers show their own solutions as well. If the students wish to get some feedback from their solutions, they are free to send the answers to the lecturer before the exercise session. In any case, it is suggested that the students try to solve the exercises themselves before the session. Or latest at the session.

**LR7 - Monday, 18th November from 11am-1pm:**Exercise Round 1**LR7 - Monday, 25th November from 11am-1pm:**Exercise Round 2**LR7 - Monday, 2nd December from 11am-1pm:**Exercise Round 3

## Course Material:

The primary materials of the course are the following (all available in PDF format through the links below):

- The first 4 lectures are based on the these
lecture notes:
- Särkkä (2012): Applied Stochastic Differential Equations. Lecture notes. (Chapters 1-5).

- The fifth lecture is based on these material:
- Särkkä (2013): Bayesian Filtering and Smoothing. Cambridge University Press.
- Särkkä (2006): Recursive Bayesian Inference on Stochastic Differential Equations. Doctoral dissertation, Helsinki University of Technology.
- Särkkä & Sarmavuori (2013). Gaussian Filtering and Smoothing for Continuous-Discrete Dynamic Systems. Signal Processing, Volume 93. Issue 2, Pages 500-510.

- The sixth lecture is based on these material:
- Rasmussen, Williams (2006): Gaussian Processes for Machine Learning. MIT Press.
- Hartikainen, Särkkä (2010). Kalman Filtering and Smoothing Solutions to Temporal Gaussian Process Regression Models. Proceedings of MLSP.
- Särkkä, Solin, Hartikainen (2013): Spatio-Temporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing. IEEE Signal Processing Magazine, Volume 30, Issue 4, Pages 51-61.

Students might find the following SDE books useful as well:

- Gardiner (2009): Stochastic Methods: A Handbook for the Natural and Social Sciences, Springer.
- Oksendal (2010): Stochastic Differential Equations: An Introduction with Applications. Springer.
- Kloeden, Platen (1992): Numerical Solution of Stochastic Differential Equations. Springer.