Stat 206A: Gibbs Measures, Fall 2006
In this course we will study the mathematical theory of Gibbs
measures with special emphasis on its recent applications in machine learning,
satisfiability, coding and computational biology. The mathematical theory of
Gibbs measures uses discrete mathematics, convexity, ergodic theory, probability
and analysis. The course will intertwine the theory with various applications.
The plan is to focus on the following topics:
1. Soft introduction to ensambles of factor graphs (chapters
912 in Constraint
Satisfaction Networks in Physics and Computation).
2. Low Density Parity Check Codes (chapters 23 in Modern Coding Theory).
3. Gibbs measures on trees and their role in studying Gibbs
measures on general graphs (papers 15)
4. Gibbs measures on trees and Phylogenetics (papers
68).
5. Additional topics and students projects. .
Prerequisites: Strong
background in probability and analysis. Background in Markov Chain Mote Carlo is
recommended. For students with no background in MCMC it is recommended to
concurrently take Sinclair's course.
One year of Stat 205 or Stat 204 is required.
Students who have not taken Stat 205 or Stat 204 and students who feel
they may not have sufficient mathematical background are encouraged to contact
the instructor in order to verify that they have the appropriate background.
Lectures: Tuesday & Thursday, 56:30pm, Evans 330.
Instructor: Elchanan Mossel (mossel@stat dot berkeley dot edu)
Grade: The grade will be based on the following:
 Participation.
 Scribing two lectures. You can find a latex file here.
 Submitting homework problems (10 points in total)
 Presenting a paper / project at the end of the term.
Bibliography  Books
 Georgii,
HansOtto Gibbs measures and phase transitions. de
Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co.,
Berlin, 1988. xiv+525 pp.
 Liggett,
Thomas M. Interacting particle systems. Reprint of the 1985 original. Classics
in Mathematics. SpringerVerlag, Berlin, 2005. xvi+496 pp
 Liggett,
Thomas M. Stochastic interacting systems: contact, voter and exclusion
processes. Grundlehren
der Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], 324. SpringerVerlag, Berlin, 1999. xii+332
 Modern Coding Theory by
T. Richardson and R. Urbanke to appear in Cambridge University Press.
 Constraint
Satisfaction Networks in Physics and Computation by M. Mezard and A.
Montanari, to appear in Oxford University Press.
Bibliography 
Papers
 Broadcasting on trees and the Ising
model. (W. Evans, C. Kenyon, Y. Peres and L. Schulman).
Ann. Appl. Probab. 10, (2000), 410433.

Information
flow on trees Mossel,
Elchanan,
Peres, Yuval Ann.
Appl. Probab. 13 no.3 817844. (2003)

The
KestenStigum Reconstruction Bound Is Tight for Roughly Symmetric Binary
Channels
Christian
Borgs,
Jennifer Chayes,
Elchanan Mossel,
Sebastien Roch
To Appear in Proceedings of FOCS 2006 (2006)
 Loopy belief
propagation and Gibbs measures. S. Tatikonda and M. I. Jordan. In D.
Koller and A. Darwiche (Eds)., Uncertainty in Artificial Intelligence
(UAI), Proceedings of the Eighteenth Conference, 2002.
 Counting
independent sets up to the tree threshold in STOC 2006 by D.
Weitz.

On
the impossibility of reconstructing ancestral data and phylogenies
Elchanan
Mossel Jour. Comput. Bio. 10 no.5 669678. (2003)

Phase transitions in Phylogeny
Elchanan
Mossel
Trans. Amer. Math. Soc. 356 no.6 23792404 (electronic).

Optimal
Phylogenetic Reconstruction Constantinos
Daskalakis,
Elchanan Mossel,
Sebatien Roch In Proceedings of the thirtyeighth annual ACM symposium
on Theory of computing (STOC 2006), 159168,. (2006)

SCRIBE
NOTES:
Scribe notes :
Motivating Example, Aug
29
Markov Factorization, Markov Property, Clifford Hammesrseley, Aug
31
Scribe notes :
Ensembles of Factor GraphsSep
5 Random 1SAT and 2SAT, Sep
7
Scribe
notes :
REM (Andrea Montanari), Sep
12 Thresholds for Random 2SAT, 2XORSAT Sep
14
Scribe
notes :
Some more SAT bounds, Sep
19 Unit Clause Propagation, Sep
21
Scribe
notes :
Second momment for SAT, Sep
26 Introduction to LDPC, Sep
28
Scribe notes :
Weight Enumerator, Regular Codes, Oct3 The Saddle Point Method and Weight Enumeration for general Codes, Oct 5
Scribe notes :
Rate and Distance Properties of LDPC, Oct
10 Capacity of LDPC Codes, Oct 12
Scribe notes :
Bit Flipping Algorithms and Expansion, Oct
17 Decay of Correlation Notions, Oct 19
Scribe notes :
Belief Propagation, Oct 24 The computation tree, Oct 26
Scribe notes :
Uniqueness of Ising on the tree, Oct 31
SUGGESTED FINAL PROJECTS (Projects suggested by
students are more than welcome!)
1. Clifford Hammersley for models
with hardcore constraints:
Geometry of rank tests,
Three Counterexamples on
Semigraphoids
2. Random subgraphs:
Percolation
on finite graphs and isoperimetric
inequalities  taken (Nov 30, Dey)
3.
Models of
Random Regular Graphs  taken (Stauffer / Ding  Nov 14) .
4. Find
simple proof for the threshold for RANDOM2SAT.
5. Hardness of sampling
solutions of 2SAT formula:
On counting independent
sets in sparse graphs.  taken (Fabrikant, Nov 9)
6.
Exponential algorithms for solving SAT problems: Find 1.999^m algorithm for
solving a general SAT formula (m = # of clauses).
7. Aging in the Random
Energy model and other models:
Dynamics of Trap Models
and
The Arcsine Law.
8.
The
differential Equations method. (+ Stochastic corrections?)
9. 2nd moment
method for vatiants of the satisfiability problem.
10.
Efficient Erasure
Correcting Codes.  Taken (Aminzadeh, Nov 28)
11.
Random Vectors of
Bounded Weight and their linear independence.
12. Any advanced topic in
coding from
Modern Coding
Theory
13. The Saddle Point Method in
Analytic
Combinatorics.
14. The laplacian and Belief Propagation:
On
the properties of Bethe Approximation and Loopy Belief Propagation on binary
networks.
15..
Bound
Propagation.  taken (Nov 28, Simma)
16.
Aldous' Objective
Method.
17.
Robust Uniqueness
and
Reconstruction.
18. Uniqueness as a local condition and on infinite systems from "Gibbs
Measures and Phase Transitions" by Georgii.
19. Uniqueness and
Reconstruction for SAT problems on trees.
20. Do spinglasses care about
geometry?
Newman's view.
21.
Planted
SAT is easy. (and relation to real life problems?)
22.
Some
easy Phase transitions in Phylogeny. (Average branch length bounds?  taken
Mefford, Nov 16)
23. I
mplementation
of Phylogenetic bounds.
24. Reconstructing MarkovRandomFields from
observations?
25. Asymptotic convex geomery of tree colorings measures.
Final Projects
1. Nov 2nd: Roch: Reconstruction
on Trees
2. Nov 7th: Daskalakis: Phylogenetic Reconstruction.
3.
Nov 9'th: Fabrikant : Hardness of sampling 2SAT solutions +
Dimakis: Random Games.
4. Nov 14th: Ding + Stauffer: Random Regular
Graphs.
5. Nov 16th: Mefford: Easy Phylogenetic bounds + Sly:
Uniqueness on graphs and Trees.
6. Nov 28th: Aminzadeh:
Efficient Erasure Correcting Codes + Simma: Bound
Propagation.
7. Nov 30 : Dey: Percolation on finite graphs +
Etesami: SAT refutations.
8. Dec 5 : Moorea
TBA + Bresler: Aldous objective method.
9. Dec 7: :
Sen: Easy proof for the 2SAT threshold + concluding remaks.