## Description

In statistics, we estimate/reconstruct objects from data. Given a noisy image for example, the aim is to reconstruct the underlying true image.

A first course in statistics typically deals with reconstruction of finite dimensional parameters, such as the mean or the standard deviation of a distribution. For many interesting applications, however, we want to assume as little as possible about the true underlying objects. Taking a fixed number of parameters is then not appropriate. Instead this should be modelled by assuming a high-dimensional or even infinite dimensional parameter space. To reconstruct an image, for instance, we can think of it as a two-dimensional function and take as a parameter space a function class.

The mathematical theory of complex statistical models has been developed largely during the past years but remains a topic of active research with many challenging open problems. One of the nice features is that there is a notion of optimality and estimators (reconstruction methods) can be constructed that (nearly) achieve this optimal behaviour.

## Course objectives

In the course we give a mathematical introduction to this field. The course is based on lecture notes that will be made available after the lectures. We start with a short introduction of mathematical prerequisites. We then discuss general estimation methods and derive rate optimal bounds for the statistical estimation risk. To illustrate the mathematical theory we discuss applications in biology, image reconstruction and finance. At the end of the course we discuss very recent developments in formulating a theory for deep neural networks.

## References

Tsybakov, A.: Introduction to nonparametric statistics. Springer, 2009.

available from: http://link.springer.com/book/10.1007%2Fb13794

Johnstone, I.: Gaussian estimation: Sequence and wavelet models. Lecture notes.

available from: http://statweb.stanford.edu/~imj/GE06-11-13.pdf

## Prerequisites

The course requires tools from various areas in mathematics such as measure theory and function spaces. We briefly introduce these concepts at the beginning of the course. As we will otherwise not be able to cover interesting theory, the idea is to discuss some of these underlying concepts in less depth. All required tools are also described in the lecture notes. An introduction to mathematical statistics, measure theory and functional analysis will therefore be very helpful but is not required.

## Assessment method

Weekly homework assignments with math problems (1/3) and a final exam (2/3).

Depending on the number of students the final exam will be oral or written.

## Registration

Via blackboard.

To be able to obtain a grade and the ECTS for the course, sign up for the (re-)exam in uSis ten calendar days before the actual (re-)exam will take place. Note, the student is expected to participate actively in all activities of the program and therefore uses and registers for the first exam opportunity.

Exchange and Study Abroad students, please see the Prospective students website for information on how to apply.

## Contact information

In case of further question, please contact directly

schmidthieber@math.leidenuniv.nl

The course is also open to 3rd year bachelor students.