# ST407: Monte Carlo Methods

###### Lecturer(s)

###### Dr Adam Johansen

* Commitment: *3 lectures per week, 1 computer practical per week starting in week 2.

** Aims: **This module will provide students with the tools for advanced statistical modelling and associated estimation procedures based on computer-intensive methods known as Monte Carlo techniques.

** Content: **When modelling real world phenomena statisticians are often confronted with the following dilemma: should we choose a standard model that is easy to compute with or use a more realistic model that is not amenable to analytic computations such as determining means and p-values. We are faced with such choice in a vast variety of application areas, some of which we will encounter in this module. These include financial models, genetics, polymer simulation, target tracking, statistical image analysis and missing data problems. With the advent of modern computer technology we are no longer restricted to standard models as we can use simulation-based inference. Essentially we replace analytic computation with sampling of probability models and statistical estimation. In this module we discuss a variety of such methods, their advantages, disadvantages, strengths and pitfalls.

**Learning Outcomes:**

- Knowledge of a collection of simulation methods including Markov chain Monte Carlo (MCMC); understanding of Monte Carlo procedures.
- Ability to develop and implement an MCMC algorithm for a given probability distribution
- Ability to evaluate a stochastic simulation algorithm with respect to both its efficiency and the validity of the inference results produced by it.
- Ability to use Monte Carlo methods for scientific applications.

*Desirable Background***:**

- A basic knowledge of the statistical programming language R or SPLUS. Coursework will be based on R.

- Probability A & B or equivalent.
- ST218/ST219 Mathematical Statistics A & B or equivalent.

*Commitment** : *30 hourly lectures and 9 hourly practicals. This module runs in Term 1.

*Assessment** : *20% by coursework (assignment 1 10%, assignment 2 10%) and 80% by exam in April.

*Syllabus**:*

1. **Introduction and Examples:** The need for Monte Carlo Techniques; history; example applications.

2. **Basic Simulation Principles:** Rejection method; variance reduction; importance sampling.

3. **Markov chain theory:** convergence of Markov chains; detailed balance; limit theorems.

4. **Basic MCMC algorithms:** Metropolis-Hastings algorithm; Gibbs sampling.

5. **Implementational issues:** Burn In; Convergence diagnostics, Monte Carlo error.

6. **More advanced algorithms:** Auxiliary variable methods; simulated and parallel tempering; simulated annealing; reversible jump MCMC.

**Books**:

- C.P.Robert and G.Casella, Monte Carlo Statistical Methods (2nd Ed.), Springer, 2004.

- J. Voss "An introduction to Statistical Computing: A Simulation-Based Approach"
- J.S. Liu, Monte Carlo Strategies in Scientific Computing, Springer, 2001.

* Deadline:* Assignment 1: Thursday of week 5. Assignment 2: Thursday of week 10.

* Feedback:* Feedback on assignments will be returned after 2 weeks, following submission.