CS774. Markov Random Field : Theory and Application Lecture 15 Kyomin Jung KAIST Oct

Sampling What is sampling?  Given a probability distribution, pick a point according to.  e.g. Monte Carlo method for integration Choose numbers uniformly at random from the integration domain, and sum up the value of f at those points  Sometimes we don’t know but can evaluate its ratios at given points (ex MRF)

How to use Sampling? Volume computation in Eucliden space. In MRF set up, one can compute marginal probabilities from random samplings. MRF on G

Some side questions In most randomized algorithm (ex sampling), we assume that we can choose uniform random numbers from {0,1} for polynomially many time. In practice, we usually use only “one time” random seed from a time function.  Pure random binary sequence is expensive. An interesting research topic called “Pseudo- random generator” deals with it.

The Gibbs Sampling Develpoed by Geman & Geman 1984, Gelfand & Smith with distribution Consider a random vector Suppose that the full set of conditional distributions where MRF

The Gibbs Sampling We assume that these conditional distributions can be sampled (in MRF, it is possible). Start at some value  The algorithm: Sample from

The Gibbs Sampling Cycle through the components again: … up to. At time n, update the i th component by from

Markov Chain  A chain of events whose transitions occur at discrete times States S 1, S 2, … X t is the state that the chain is in at time t Conditional probability  P(X t =S j |X t1 =S i1, X t2 =S i2, …, X tn =S in )  The system is a Markov Chain if the distribution of Xt is independent of all previous states except for its immediate predecessor Xt-1  P(X t =S j |X 1 =S i1, X 2 =S i2, …, X t-1 =S it-1 )=P(X t =S j |X t-1 =S it-1 )

Stationary distribution Stationary Distribution of Markov Chain  If Markov Chain satisfies some mild conditions, it gradually forget its initial state.  Eventually converge to a unique stationary distribution. Gibbs Sampling satisfies Detailed Balance Equation  So Gibbs Sampling has as its stationary distribution. Hence, Gibbs Sampling is a special case of MCMC (Mrokov Chain Monte Carlo method).

An MRF on a graph G exhibits a correlation decay (long-range independence), if when is large Remind: Correlation Decay Practically, Gibbs Sampling works well (converges fast) when the MRF has correlation decay.

A similar algorithm: Hit and Run Hit and Run algorithm is used to sample from a convex set in an n-dimensional Eucliden space. It converges in