Section outline
-
The module introduces learning in probabilistic models. We will discuss fundamental algoritms and concepts, including Expectation-Maximization, sampling and variational approximations, and we will study relevant models from the three fundamental paradigms of probabilistic learning, namely Bayesian networks, Markov networks and dynamic models. Models covered include: Hidden Markov Models, Markov Random Fields, Boltzmann Machines, Latent topic models.
Date Topic References Additional Material 7 04/03/2026
(16-18)Learning with fully observed variables
learning as inference; flavors of probabilistic learning; Maximum Likelihood learning with fully observed variables; Naïve Bayes
[BRML] Sect. 9.1.1-9.1.1.3, 9.3, 10.1, 10.2 A dedicated chapter to deepen knowledge on fitting distributions by ML or MAP. 8 05/03/2026
(11-13)Learning with hidden variables
Latent/hidden variable models, maximum likelihood learning with latent variables;
Expectation-Maximization algorithm; exact learning in mixture models[BRML] Sect. 11.1 (learning with latent variables)
[BRML] 20.1, 20.2.1, 20.3 (mixture models)
9 10/03/2026
(11-13)Hidden Markov Models - Part I
lgenerative models for sequential data; inference problems on sequential data; forward-backward algorithm;[BRML] Sect. 23.1.0 (Markov Models)
[BRML] Sect. 23.2.0-23.2.4 (HMM and forward backward)
Additional Readings
[1] A classical tutorial introduction to HMMs10 11/03/2026
(16-18)Hidden Markov Models - Part II
EM learning in HMMs; Viterbi algorithm; advanced models
[BRML] Sect. 23.3.1-23.3.4 (EM and learning)
[BRML] Sect. 23.2.6 (Viterbi)
Software
HMMLearn - Scikit-like library for HMMs
HMMS - discrete and continuous time HMMs
11 12/03/2026
(11-13)Variational Inference
learning and inference in intractable latent variable models; expectation lower-bound; generalized expectation maximization
BRML] Sect. 11.2.1 (Variational EM)
12 17/03/2026
(11-13)Latent Dirichlet Allocation (LDA)
latent topic models; probabilities as random variables; Dirichlet distribution; LDA learning by variational inference; LDA applications
[BRML] Sect. 20.4-20.6.1 (LDA)
Additional Readings
[2] LDA foundation paper
[3] A gentle introduction to latent topic models
Sofware
- A tutorial implementation of LDA in R that does not use any high-level API
- A step-by-step tutorial on using LDA for text topic modelling in Gensim
- A chatty demo on using LDA for image understanding
13 18/03/2026
(16-18)Sampling methods
sampling fundamentals; ancestral sampling; Gibbs Sampling; approximated LDA parameter learning via sampling
[BRML] Sect. 27.1 (sampling), Sect. 27.2 (ancestral sampling), Sect. 27.3 (Gibbs sampling)
Additional Readings
[4] A step-by-step derivation of collapsed Gibbs sampling for LDA14 19/03/2026
(11-13)Markov Random Fields
learning in undirected graphical models; conditional random fields; restricted Boltzmann machine; contrastive divergence and Gibbs sampling in use
[BRML] Sect. 4.2.2, 4.2.5 (MRF)
[BRML] Sect. 4.4 (Factor Graphs)
[BRML] Sect. 5.1.1 (Variable Elimination and Inference on Chain)
[BRML] Sect. 9.6.0, 9.6.1, 9.6.4, 9.6.5 (Learning in MRF/CRF)
Additional Readings
[5] A clean and clear introduction to RBM from its author