MONTE CARLO AND EMPIRICAL METHODS FOR STOCHASTIC INFERENCE | FMS091 |

**Aim**

The purpose of the course is to give the students tools and knowledge to, given a data material and a stochastic model description with a problem formulation, be able to identify and formulate relevant quantities, and approximate these quantities numerically using Monte Carlo based methods. The main aim lies in enhancing the scope of statistical problems that the student will be able to solve.

*Knowledge and understanding*

For a passing grade the student must

- be able to explain and use the concept of statistical uncertainty from a frequentist perspective as well as from a Bayesian perspective,
- be able to describe fundamental principles of random variable generation and Monte Carlo integration.

*Skills and abilities*

For a passing grade the student must

- given a stochastic model and problem formulation, be able to choose relevant quantities in a way that permits approximation using Monte Carlo methods,
- given a (possibly multivariate) probability distribution, suggest and implement in a computer program, a method for generation of random variables from this distribution,
- given a large number of generated random variables from a probability distribution, approximate relevant probabilities and expectations as well as estimate the uncertainty in the approximated quantities,
- present the course of action taken and conclusions drawn in the solution of a given statistical problem.

*Judgement and approach*

For a passing grade the student must

- be able to identify and problemize the possibilities and limitations of statistical inference.

**Contents**

Simulation based methods of statistical analysis. Markov chain methods for complex problems, e.g. Gibbs sampling and the Metropolis-Hastings algorithm. Bayesian modelling and inference. The re-sampling principle, both non-parametric and parametric. The Jack-knife method of variance estimation. Methods for constructing confidence intervals using re-sampling. Re-sampling in regression. Permutations test as an alternative to both asymptotic parametric tests and to full re-sampling. Examples of mor complicated situations. Effective numerical calculations in re-sampling. The EM-algorithm for estimation in partially observed models.

**Literature**

Sköld, M: Computer Intensive Statistical Methods, Avd. för Matematisk statistik, LU, 2006.