## Markov Random Fields

#### March-May 2014

During March-May 2014, I (Olle Häggström) will be giving a Ph.D. course on Markov random fields at the Department of Mathematical Sciences at Chalmers/GU.

The Markov random field concept extends the Markov property from the basic chain structure of discrete time Markov chains to spatial and other network structures, and has turned out to be highly fruitful in as diverse areas as image analysis and statistical mechanics. This course treats probabilistic and statistical aspects of Markov random fields, including dependence structure, phase transitions, simulation and parameter estimation. Particular attention will be paid to concrete examples such as the Ising and Potts models, and Gaussian Markov random fields.

Literature: The course will is based mostly on (parts of) the book Image Analysis, Random Fields and Markov Chain Monte Carlo Methods by Gerhard Winkler, the paper The random geometry of equilibrium phases by Georgii, Häggström and Maes (2001). Brief lecture notes will be posted here later (final version no later than May 20).

A preliminary list (subject to changes until May 16) of The material you need to know for the exam is this:

On a detailed level (being able to carry thorugh proofs etc): On a more cursory level (some familiarity with concepts, etc):
• Winkler, Chapters 4, 5, plus the chapters, enumerated differently in different editions of the book, entitled Metropolis Algorithms, Maximum likelihood Estimation and (Consistency of) Spatial ML estimators.
• GHM, Chapters 2, 3 and 4.
Schedule:
 Wednesday, March 19 08.00-09.45 MVL14 Basic MRF concepts and definitions, loosely based on Winkler, Sections 3.1 and 3.2. Will try to sell you my own brand-new terminology distinguishing between local, regional and global MRF properties. And the Ising model. (Plus, I was tricked into misnaming the ∂ symbol.) Wednesday, March 26 08.00-09.45 MVL14 The Hammersley-Clifford Theorem, relating Gibbs and Markov random fields, following Winkler, Section 3.3. (Plus, will revert to the standard name of the ∂ symbol.) Friday, March 28 10.00-11.45 MVL14 When are the local and regional MRF properties equivalent? I'll prove (using coupling of Markov chains - one of my favorite tools, and you'll see more of it as the course moves on) that under Winkler's positivity condition they are. And I'll show a conterexample to demonstrate that without that condition they're not. Friday, April 4 10.00-11.45 MVL14 I'll begin discussing the Ising model on Zd based on Chapters 2-4 in GHM (2001). Wednesday, April 9 08.00-09.45 MVL14 Holley's Theorem (Thm 4.8 in GHM (2001)) and some of its ramifications. Monday, April 14 (Note: change of day!) 10.00-11.45 MVL14 The phase transition phenomenon for the Ising model on Z2: proof of Gibbsian uniqueness for β close to 0, and Gibbsian nonuniqueness for β large. Wednesday, April 16 08.00-09.45 MVL14 The FKG inequality (Thm 4.11 in GHM (2001)). The inhomogeneous Ising model on Z1, ruthlessly exploited as an infinite-volume counterexample to FKG and to the global Markov property. Friday, April 25 10.00-11.45 MVL14 MCMC for Markov random fields, very loosely based on Chapter(s 4 and) 5 in Winkler. Wednesday, April 30 08.00-09.45 MVL14 For how long do we need to run an MCMC chain? Suitable (i.e., more uselful) substitutes for Winkler's Theorems 4.3.2 and 5.1.4. Wednesday, May 7 08.00-09.45 MVL14 Maximum likelihood estimation in Markov random fields, digging into Chapters 13 and 14 of Winkler. Friday, March 9 10.00-11.45 MVL14 The Ising model with external field, or in other words with the parameter h in the displayed equation on p 14 of GHM (2001) (before today h has been set to 0). My treatment of this topic comes straight from my heart, and has no counterpart in the existing literature. Friday, May 16 10.00-11.45 MVL14 Briefly about the Potts model and Gaussian Markov random fields.

The exam takes place on Wednesday, May 28 at 8:30-11:30. in rooms MVL22 and MVL23. It will be a combination of written and oral examination. (More or less, the oral part will be your chance to defend your written solutions.) Successful completion of the exam will be rewarded by 7.5 hp credit points.

Mailing list: Contact me at olleh@chalmers.se if you wish to be kept posted with course information.