Probabilistic Graphical Models

Directed (Bayesian networks) vs. undirected (Markov Random Fields) models, Factorization of joint distributions, Conditional independence and d-separation, Markov properties and Hammersley-Clifford theorem, Graphical model representation advantages, Exponential family distributions and sufficiency, Common probability distributions in graphical models.

Exact inference (variable elimination, belief propagation), Complexity analysis of inference, Approximate inference (importance sampling, likelihood weighting), Markov Chain Monte Carlo (MCMC) methods (Metropolis-Hastings, Gibbs sampling), Sequential Monte Carlo (particle filters), Loopy belief propagation and mean field approximation.

Parameter learning in Bayesian networks (maximum likelihood, MAP estimation), Dirichlet priors and Dirichlet-multinomial models, Structure learning (score-based, constraint-based), K2 algorithm and greedy hill-climbing, Markov blanket discovery, PC algorithm for DAG structure learning, Learning Markov Random Fields (pseudolikelihood estimation).

Hidden Markov Models (HMM) - three fundamental problems (evaluation, decoding, learning), Forward-backward algorithm, Viterbi algorithm, Baum-Welch EM algorithm, Dynamic Bayesian Networks (DBN) for time series, Switching Kalman filters, Continuous-time graphical models.

Gaussian graphical models and precision matrix estimation, Conditional Random Fields (CRF) for structured prediction, Factor graphs and sum-product algorithm, Nonparametric Bayesian models (Indian Buffet Process), Deep generative models with graphical structure, Causal discovery from observational data, Applications in bioinformatics, NLP, and robotics.