Conditional independence, Markov properties and graphoid axioms. Hammersley-Clifford theorem, exponential family and canonical parameters, decomposable graphical models and criteria of decomposability.
Gaussian graphical models (GGM), covariance and concentrations graph models, Bayesian parametric inference on GGMs, a family of hyper- Wishart distributions on decomposable GGMs, model determination in GGMs. Discrete hierarchical log-linear models, Bayesian analysis on graphs for contingency tables, a family of hyper-Dirichlet distributions.
Sampling algorithms for both graph and parametric posterior inference.
Project work comprises graph modelling and analysis where theoretical knowledge acquiring during the course will be applied within a chosen area of interests.
After having passed the course the student is supposed to be able to:
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State the Hammersley-Clifford theorem for undirected graphs and explain its connection to the factorization of the underlying probability distribution;
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State the graphoid axioms and relate them to the dependence structure induced by graph separation and conditional independence in a multivariate probability distribution;
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Derive the fundamentals of the Gaussian graphical models and log-linear modes for contingency tables;
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Derive the concept of decomposability of a graph and explain its role for both graph structure learning and parametric inference;
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Explain the role of hyper-Wishart and the hyper-inverse Wishart distributions for the Bayesian inference within the context of Gaussian graphical models;
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Explain the role of hyper-Dirichlet distribution for Bayesian inference within the context of the log-linear modes for contingency tables;
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Judge whether probabilistic graphical modelling can be regarded as a promising inferential strategy for a given real-world problem;
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Design, apply and validate a graph structure learning algorithms along with the corresponding parametric inference strategy, suitable for a specific real-world consideration.
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Place the probabilistic graphical modelling into a general perspective of multivariate statistical inference.
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Review the modern literature on a selected topic of the probabilistic graphical modelling and write a technical report presenting graph-theoretic concepts and algorithms.
