The module introduces decision theory, hypothesis testing, "Monte Carlo" simulation, Bayesian inference, comparative inference and the generalised linear model.
On completion of the course students should be able to (learning outcomes):
Syllabus:
Decision theory
Loss, risk, admissible and inadmissible decisions, randomised decisions. Minimax decisions and Bayes' solutions, including simple results.
Hypothesis testing
Simple and composite hypotheses, types of error, power, operating characteristic curves, p-value. Neyman-Pearson method. Generalised likelihood ratio test.
Use of asymptotic results to construct tests. Central limit theorem, asymptotic distributions of maximum likelihood estimator and generalised likelihood ratio test statistic.
Bayesian inference
Prior and posterior distributions. Choice of prior: bets, conjugate families of distributions, vague and improper priors. Predictive distributions. Bayesian estimates and intervals for parameters and predictions. Bayes factors and implications for hypothesis tests. Use of Monte Carlo simulation of the posterior distribution to draw inferences. Bayesian and Empirical Bayes approach to credibility theory and use it to derive credibility premiums in simple cases.
Comparative inference
Different criteria for choosing good estimators, tests and confidence intervals. Different approaches to inference, including classical, Bayesian and non-parametric.
Generalised linear model
Explain the fundamental concepts of a generalised linear model (GLM), and describe how a GLM may apply.
On completion of the course students should be able to (learning outcomes):
- Understand concepts of decision theory;
- Understand hypothesis testing, exact and asymptotic tests, properties of tests;
- Understand basic principles of Bayesian inference;
- Understand principles and methods to choose good estimators
- Understand basic concepts of a generalised linear model.
Syllabus:
Decision theory
Loss, risk, admissible and inadmissible decisions, randomised decisions. Minimax decisions and Bayes' solutions, including simple results.
Hypothesis testing
Simple and composite hypotheses, types of error, power, operating characteristic curves, p-value. Neyman-Pearson method. Generalised likelihood ratio test.
Use of asymptotic results to construct tests. Central limit theorem, asymptotic distributions of maximum likelihood estimator and generalised likelihood ratio test statistic.
Bayesian inference
Prior and posterior distributions. Choice of prior: bets, conjugate families of distributions, vague and improper priors. Predictive distributions. Bayesian estimates and intervals for parameters and predictions. Bayes factors and implications for hypothesis tests. Use of Monte Carlo simulation of the posterior distribution to draw inferences. Bayesian and Empirical Bayes approach to credibility theory and use it to derive credibility premiums in simple cases.
Comparative inference
Different criteria for choosing good estimators, tests and confidence intervals. Different approaches to inference, including classical, Bayesian and non-parametric.
Generalised linear model
Explain the fundamental concepts of a generalised linear model (GLM), and describe how a GLM may apply.
- Module Supervisor: Yanchun Bao