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)

Date
 		Lesson topics
 		links
24/02/2025
 Bayes theorem. MAP estimates. Elementary examples of inference with Bayes theorem. Meaning of Bayesian inference. The approach of R.T. Cox to the logic of approximate reasoning.
25/03/2025 The approach of R.T. Cox to the logic of approximate reasoning (ctd.)
Example of Bayesian inference: parameter of a binomial model (and Beta pdf posterior). Cromwell's rule (Lindley). Bayesian credible intervals. Conjugate priors. Connection with frequentist statistics.
26/03/2025 Multinomial distributions and the Dirichlet pdf. The multivariate Gaussian distribution. "Completing the square", conditional Gaussian distributions and marginalized Gaussian distributions.
Determination of a scale factor for experimental uncertainties with Bayes theorem and Jeffreys' priors.
27/03/2025 Introduction to objective priors. Bartlett identities. Cramér-Rao Bound. Information-theoretic concepts in statistics. The Kullback-Leibler divergence. Jeffreys' priors.
07/04/2025 Bernardo's reference priors.
Edwin Jaynes and the Maximum Entropy (MaxEnt) principle. 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.
Introduction to model selection.
08/04/2025 Model selection (ctd.)
Monte Carlo methods in the Bayesian approach, part 1: 1. Review of acceptance-rejection sampling; 2. importance sampling; 3. statistical bootstrap. 4. Bayesian methods in a sampling-resampling perspective.
09/04/2025 Monte Carlo methods in the Bayesian approach, part 2: 4. Bayesian methods in a sampling-resampling perspective. 5. introduction to Markov chains. 6. Detailed balance and Boltzmann's H-theorem. 7. The Gibbs sampler.
10/04/2025 8. More on Gibbs sampling. 9. Simulated annealing and the Traveling Salesman Problem (TSP). 10. The Metropolis algorithm. 11. Image restoration and Markov Random Fields (MRF)
05/05/2025
06/05/2025
07/05/2025
08/05/2025

Freely available study books