# ST414: Advanced Topics in Statistics

###### Lecturer(s)

##### ∗∗∗ Please note that this module will not be running in the 2018/19 academic year ∗∗∗

*Prerequisite(s)**:* ST218/219 Mathematical Statistics A&B

** Content**: Three self-contained sets of ten lectures. This module runs in Term 2.

**Rough Paths and Statistics - Dr Anastasia Papavasiliou**

**Aims:**

Understand and learn how to use some of the basic concepts of the theory of rough paths, in particular in terms of data description.

**Content:**

The theory of rough paths is a relatively new theory, aiming to describe "rough" paths and the way they interact with systems. We will go through the basic concepts and ideas and, towards the end, read about some recent applications in statistics.

**Recommended:**

Some understanding of Brownian motion (an example of a rough paths) and differential equations will be useful.

**Stein-Chen Method for Poisson Approximations - Dr Zorana Lazic
**

*Aims: *

Stein-Chen method is a powerful modern technique which extends the Poisson “law of small numbers” (given n independent events each of small probability p, the total number of event which occur is approximately Poisson of mean np). The course will cover the fundamentals of Stein-Chen method to illustrate the construction of the Stein equation and the derivation of the properties required on its solution. A number of coupling methods for use in the Stein equation will be presented, as well as its use in cases of local dependence along with applications in various disciplines.

*Objectives:* By the end of the course students will be able to:

1. Understand and describe the principles of the Stein-Chen method.

2. Apply the Stein-Chen method to examples.

*Recommended:*

Basic knowledge of Probability Theory (Expectation, independence, conditional distributions, Bernoulli indicator random variables. Poisson random variables). The course ST111 (Probability A) covers all of them and is essential for this course. ST318 (Probability Theory) might also help with various proofs and ideas, but is not mandatory.

The course will be motivated by examples from many applications including engineering, economics and biological sciences.

**Approximate Bayesian computation - Prof Christian Robert
**

**Description: **

When facing complex probabilistic models, it sometimes happens that the resulting distributions are intractable in the sense that the corresponding densities cannot be computed for a given value of the parameter and a given value of the parameter. Doing Bayesian analysis on such models then becomes a challenge since traditional techniques fail to apply.

There are several ways of dealing with this intractability problem by Monte Carlo techniques and we will consider a collection of methods called approximate Bayesian computation (ABC) that replace computing the intractable density by multiple simulations from this density. The evaluation, calibration and implementation of this method will be central to this course. We will also provide entries on Bayesian inference, Monte Carlo methods and indirect inference, as well as illustrations in population genetics (Kingman's coalescent).

**Recommended:**

Some familiarity with basic Monte Carlo methods (in particular Rejection Sampling), Markov Chains and SDEs would be useful (although certainly not essential).