Markov random fields and applications (series of lectures)
A talk in the Cluster Group Stochastic Analysis series by
Tetyana Pasurek
| Abstract: | Markov fields appear as Gibbs equilibrium states in statistical
mechanics or as Markov networks or undirected graphical models in Big
Data analysis and machine learning. The goal is to develop a unified
theory that can then be applied to specific models (classical or
quantum; on lattices $Z^{d}$ and general graphs or in the continuum $R^{d}$), covering a large amount of the results known so far. In particular, we address the problems of existence and uniqueness of Markov fields, their mixing properties and dimension-free estimates of convergence rates. We also connect the two basic approaches - Dobrushin's theory of weak dependence and Ruelle's superstability
estimates - and extend them to unbounded interactions and irregular underlying spaces.
Lecture I. Harris' ergodic theorem: from Markov chains to Markov fields
We discuss the existence and uniqueness problems for Markov fields in the framework of Dobrushin's contraction method and its local versions. As a by-product of our approach (or as a particularly simple case), we provide an elementary and self-contained proof of Harris' ergodic theorem for Markov chains in the spirit of Hairer and Mattingly (2008). |