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 (41st PhD cycle)

(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 11. Image restoration and Markov Random Fields (MRF). 12. Introduction to the Markov Chain Monte Carlo method with an application to medical physics. 
06/05/2025 13. The efficiency of MCMC methods. 14. Affine-invariant MCMC algorithms (emcee).
Corner plots. Detailed coding example on cell survival curves (LQ model).
07/05/2025 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
08/05/2025 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.

Freely available study books


MCMC resources