Course description

This course is a basic introduction to Bayesian techniques, in the framework of the Physics PhD course of the Physics Department of the University of Trieste.

Course program

(grayed text means preliminary program)

Lesson topics
Bayes theorem. Example of inference with Bayes theorem. Examples and applications of Bayes' theorem (medical tests, etc.). Bayesian inference, discrete hypotheses, parameter inference.
Example of Bayesian inference: parameter of a binomial model (and Beta pdf posterior). Cromwell's rule (Lindley). Bayesian credible intervals. (link to slides)
20/04/2020 A decision problem (from Skilling, 1998). Analytical Bayesian straight-line fit. Discussion on prior distributions and random variable transformations. Physical models and prior distributions (physical coin toss, Bertrand's paradox). Jeffreys' priors. (link to slides)
23/04/2020 Boltzmann entropy and Shannon entropy. Edwin Jaynes and his approach to statistical mechanics. The kangaroo problem as an example of ill-posed problem and its regularization by entropy maximization. Objective priors with the maximum entropy method in both the discrete and the continuous case (using the KL divergence). Simple example of application to image restoration (Skilling). (link to slides)
24/04/2020 Examples of application of the Bayesian approach: 1. weighted mean; 2. miscalibrated Gaussian measurement errors; 3. search for weak signals in spectra; 4. expert elicitation; 5. the lost flight AF 477 and other search problems; 6. the statistical link between smoking and lung cancer. (link to slides)
30/04/2020 Monte Carlo methods in the Bayesian approach, part 1. 1. Review of the acceptance-rejection sampling; 2. importance sampling; 3. statistical bootstrap; 4. Bayesian methods in a sampling-resampling perspective; 5. introduction to Markov chains. (link to slides)
01/05/2020 Monte Carlo methods in the Bayesian approach, part 2. The Markov Chain Monte Carlo (MCMC). (link to slides). A few examples of the MCMC at work. (link to slides)
04/05/2020 Applications of Bayesian methods. 1. Image restoration; 2. Bayes and automatic classification. 3. Bayesians, frequentists and all that, a discussion on the principles of statistics. (link to slides)
05/05/2020 Applications of Bayesian methods. 4. The EM algorithm; 5. The mass distribution of neutron stars. (link to slides)

Useful links