## CS294: MARKOV CHAIN MONTE CARLO: FOUNDATIONS & APPLICATIONS, FALL 2009

INSTRUCTOR: Alistair Sinclair
(sinclair@cs)

TIME: Tuesday, Thursday 09:30-11:00

PLACE: 310 Soda

OFFICE HOURS: Monday 1:00-2:00, Tuesday 11:00-12:00 in 677 Soda

## RECENT ANNOUNCEMENTS

(11/30/09) There is a typo in Q5(b) of HW2: the n^4 should be replaced by n^3. This is
corrected in the version below. Sorry for any confusion.

(11/22/09) There was a missing reversibility condition in Q4 of HW2; this is corrected
in the version posted below, which also includes an expanded hint. Apologies for the confusion.

(11/13/09) Homework 2 (the second and final homework of the class) is posted
below. It is due Friday Dec 11 (the same day as the project reports). The long
period between posting and due date is to allow everyone to schedule their time
at the end of the semester.

(11/9/09) A schedule for project presentations is posted here.
**Please check your place on the list and inform me of any conflicts immediately!**
Presentations start one week from tomorrow, Tuesday November 17.

(10/27/09) Details of the Class Project are posted here. A
list of suggested topics can be found here. Please note
that you need to select your topic by **next Friday, November 6th!**

(10/27/09) Homework 1 Solutions are posted below. My apologies for forgetting
to post them earlier.

(10/14/09) Homework 1 has been graded and can be picked up in class.

(9/28/09) I've substituted a more interesting version of Q6 on the Problem
Set; see new version below. Please disregard the original version.

(9/28/09) There was a broken link to the first Problem Set, which is
now fixed. My apologies. As a result, the deadline is extended to Monday,
Oct 5 at 5pm. There was also a typo on Q3 which has been fixed in the
version below.

(9/23/09) The first Problem Set is posted below; it is due next
Friday (October 2nd).

(9/22/09) There will be NO CLASS on Thursday, Sept 24 in view of
events on campus that day. We will make up this class during the
project presentations at the end of the semester.

(9/16/09) Would those of you who are auditing the class (and are
Berkeley students) please officially sign up on a pass/no-pass basis
(i.e., S/U for graduate students, P/NP for undergraduates) as soon as
possible? This ensures that the department receives full credit
for offering the class. The requirement for passing the class on
this basis is simply attending lectures and helping with scribe notes.
Thank you for your cooperation.

(9/16/09) Scribe notes for the first four lectures are posted below (i.e.,
we are up to date!). Thanks to the scribes for their efforts.

(8/27/09) There will be **NO LECTURES** (and no office hours) next week
(Sept. 1 & 3). The class will continue as usual on Tuesday Sept. 8. The missing
time will be made up with project presentations at the end of the semester.

## COURSE DESCRIPTION

The Markov chain Monte Carlo method (MCMC) is
a powerful algorithmic paradigm, with applications
in areas such as statistical physics, approximate counting,
computing volumes and integrals, and combinatorial optimization.
For many years it has been applied
without rigorous justification, so that
numerical results derived from it have to be taken on faith.
More recently, techniques have emerged for analyzing MCMC algorithms
and their applications, which allow one to derive
precise performance guarantees for them. The techniques fall into
three main domains --- probabilistic, combinatorial and geometric ---
and are of interest in their own right, having connections to such
topics as expander graphs, isoperimetric inequalities, multicommodity
flow, and spatial decay of correlations in Markov random fields.
This course will introduce these techniques and use them to
obtain provably efficient MCMC algorithms for a variety of problems.
It will also touch on some related topics, such as phase transitions
in statistical physics, simulated annealing and genetic algorithms,
and the complexity of combinatorial enumeration problems.

**Tentative List of Topics**

Basis of the Markov chain Monte Carlo (MCMC) method: ergodicity,
stationary distributions, reversibility, mixing times.
General applications of random sampling: card shuffling, approximate
counting, volume and integration, statistical physics, combinatorial
optimization.
Probabilistic methods: Stopping times, coupling,
path coupling, perfect sampling (coupling
from the past and Fill's algorithm), evolving sets. Applications:
simple shuffles, planar lattice structures, independent sets and colorings,
etc.
Multicommodity flows: Reversible Markov chains
and eigenvalues, canonical paths and flows, Diaconis-Stroock
comparison technique. Applications: monomer-dimer model and
matchings, the Ising model, 0-1 knapsack problem, etc.
Geometry: Conductance and isoperimetric inequalities,
expanders, the Cheeger bound, "average" conductance. Applications:
volume of convex bodies, contingency tables, "torpid" mixing of
Glauber dynamics.
Functional analysis: Spectral gap, log-Sobolev constant.
Spatial mixing: Uniqueness of Gibbs measures, Dobrushin's condition,
equivalence of spatial and temporal mixing. Applications: phase transitions
in statistical physics, the Ising model, reconstruction problems on trees.
Search heuristics in combinatorial optimization. The Metropolis
algorithm, simulated annealing, simulated tempering, etc.
Approximate counting: #P-completeness, fully-polynomial approximation
schemes, robustness of approximation for counting problems.

## MATERIALS

Most of the material in the class is drawn from research
papers; references will be given as we go along. Participants in
the class will also compile scribe notes, working in pairs;
each pair will be expected to act as scribe for one or two
lectures, depending on enrollment. To ensure smooth running
of the course, notes for the Tuesday and Thursday lectures are due
on the following Friday and Monday respectively. After revision,
they will be posted on the web page. The scribe notes will form
part of the assessment for the class; the credit awarded will
decrease exponentially with the time I have to spend cleaning
them up.

## ASSESSMENT

Assessment will be based on some combination of (a)~an end-of-semester
project (involving presentation of and/or incremental work on a
research paper); (b)~homeworks; and (c)~scribe notes.
This combination will depend on the enrollment and
composition of the class. Further details will be circulated
in due course.

## SCRIBE NOTES

## HOMEWORKS

## OTHER HANDOUTS