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. (link to slides)
4/05/2021 Example of Bayesian inference: parameter of a binomial model (and Beta pdf posterior). Cromwell's rule (Lindley). Bayesian credible intervals. A decision problem (from Skilling, 1998). Analytical Bayesian straight-line fit.  (link to slides)
5/05/2021 Analytical Bayesian straight-line fit (ctd.). Discussion on prior distributions and random variable transformations. Physical models and prior distributions (Bertrand's paradox). Jeffreys' priors.
Boltzmann entropy and Shannon entropy. (link to slides)
6/05/2021 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).
Examples of application of the Bayesian approach: 1. miscalibrated Gaussian measurement errors. (link to slides)
10/05/2021 Examples of application of the Bayesian approach (ctd.): 2. search for weak signals in energy spectra;  3. the statistical link between smoking and lung cancer.
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;  (link to slides)
11/05/2021 Monte Carlo methods in the Bayesian approach, part 2. 5. introduction to Markov chains. 6. The Markov Chain Monte Carlo (MCMC). (link to slides)
12/05/2021 Examples of MCMC at work (parameters of surviving fraction models in the irradiation of human cells; Bayesian line fit). MCMC software.
Applications of Bayesian methods to Image restoration (link to slides)
13/05/2021 Applications of Bayesian methods: 1. Bayes and automatic classification. 2. Logit regression. 3. Bayesian model selection. 4. The EM algorithm (with examples). (link to slides)

Useful links