Course description

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

Course program (42nd PhD cycle)

(grayed text means preliminary program)

Date
 		Lesson topics
 		links
13/04/2026
 Bayes theorem. MAP estimates. Elementary examples of inference with Bayes theorem. Meaning of Bayesian inference.
Example of Bayesian inference: parameter of a binomial model (and Beta pdf posterior). Cromwell's rule (Lindley). Bayesian credible intervals. Conjugate priors. 
14/04/2026 Connection with frequentist statistics. Multinomial distributions and the Dirichlet pdf. The multivariate Gaussian distribution. Determination of a scale factor for experimental uncertainties with Bayes theorem and Jeffreys' priors. General approach to "Completing the square".
15/04/2026 General approach to "Completing the square" (ctd.). Conditional Gaussian distributions and marginalized Gaussian distributions. A modern approach to polynomial fits. Bayesian formulation of the polynomial fit problem. Linear prediction. Introduction to model selection.
16/04/2026 General discussion on model selection: AIC, BIC, Bayes and all that, with an interlude on Wilk's theorem.
Back to the basics of Bayesian inference: Cox's proof of the logical consistency of the Bayesian method.



27/04/2026 Naive Bayesian classification. Extremely quick introduction to neural networks. Introduction to objective priors. Bartlett identities. Cramér-Rao Bound.  Information-theoretic concepts in statistics.
28/04/2026 Information-theoretic concepts in statistics. (ctd.) Kullback-Leibler divergence. Jeffreys' priors. 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.
29/04/2026 Objective priors with the maximum entropy method in both the discrete and the continuous case. (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. 4. Bayesian methods in a sampling-resampling perspective.
30/04/2026 Monte Carlo methods in the Bayesian approach, part 2: 5. introduction to Markov chains. 6. Detailed balance and Boltzmann's H-theorem.



11/05/2026 Monte Carlo methods in the Bayesian approach, part 3: 7. The Gibbs sampler.  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) 12. Introduction to the Markov Chain Monte Carlo method with an application to medical physics. 
12/05/2026 Monte Carlo methods in the Bayesian approach, part 4: 13. The efficiency of MCMC methods. 14. Affine-invariant MCMC algorithms (emcee).
Corner plots. Detailed coding example on cell survival curves (LQ model).
13/05/2026 Multiprocessing with emcee. Radiocarbon dating. Radiocarbon activity and half-life from the data reported in the 1949 paper by Willard Libby, using an emcee-based Python code

14/05/2026 The temperature record from ice core data. Estimate of the Milankovitch cycles from the ice core data with an emcee-based program. Issues with MCMC codes and list of existing codes. The role of parallel tempering in the estimate of the periods of exoplanets in the 47 UMa system.
Room allocations:

April

13  9am-11am   Phys. Dept., meeting room
14  9am-11am   Building B, meeting room
15  9am-11am   Phys. Dept., meeting room
16  11am-1pm   Phys. Dept., meeting room

27  9am-11am   Phys. Dept., meeting room
28  9am-11am   Phys. Dept., meeting room
29  9am-11am   Phys. Dept., meeting room
30  11am-1pm   Phys. Dept., meeting room

May

11  9am-11am   Building B, meeting room
12  9am-11am   Building B, meeting room
13  9am-11am   Building B, meeting room
14  11am-1pm   Building G, room V


Freely available study books


MCMC resources