Lecturer: Piotr Zwiernik (University Pompeu Fabra, Barcelona)
Latent tree models are graphical models defined on a tree, in which only a subset of variables is observed. They were first discussed by Judea Pearl as tree-decomposable distributions to generalise star-decomposable distributions such as the latent class model. Latent tree models, or their submodels, are widely used in: phylogenetic analysis, network tomography, computer vision, causal modeling, and data clustering. They also contain other well-known classes of models like hidden Markov models, Brownian motion tree model, the Ising model on a tree, and many popular models used in phylogenetics. This lecture offers a concise introduction to the theory of latent tree models. I will emphasise the role of tree metrics in the structural description of this model class, in designing learning algorithms, and in understanding fundamental limits of what and when can be learned.
This lecture course is divided into three parts. In part 1, I will present basic combinatorial concepts related to trees and tree metrics. In part 2, I will define latent tree graphical models and discuss their basic properties. I will also discuss linear latent tree models which provide a convenient general family of distributions whose second-order moment structure is tree-like. In the last part I will present main ideas used in the design of learning procedures for this model class. This includes the structural EM algorithm and various distance based methods.
This course will be based on my book
P. Zwiernik, “Semialgebraic statistics and latent tree models”, Chapman&Hall, 2015,
and a forthcoming chapter in “Handbook of Graphical Models”, see also arXiv:1708.00847.