# Proving Gibbs Distribution Implies Markov Random Field

Trivial because of exponential

Look at the handout given in class for a detailed proof.

In summary, at the end of the day, sites without s cancel.

Back to the Image Segmentation Problem

Our image model

 true image f

line process l not observed

Goal

• Given observed image g, find a probability distribution of true image f, and the line process l.

The line process estimate solves the image segmentation problem.

The true image estimate solves the image restoration problem.

Both problems are simultaneously solved!

• Note: only works for piecewise smoothe images Ž no textures

Our model assumes:

Our solution:

 prior distribution

Interesting term:

Where we assume every pixel has independent noise h ~ N ( m , s )

e.g. Poisson process noise in CCDs

Result:

is the posterior probability of a particular f, l given g. Note this is also a Gibbs distribution!

## MAP (maximum a posteriori) Estimate

If you insist on a single answer then return f*,  l* that maximizes

or equivalently, minimizes the energy function

Problem:           f , l space is very large!!

Solution:           Construct samples of f , l in this space with high probablity

Technique:        Markov Chain Monte Carlo (MCMC) lets you sample the posterior distribution

## Sampling a Distribution

Q:        How do we represent a probability distribution with sampling?

A:         Create many samples drawn from that distribution and count!

Example:

Q:        P(X > 17) = ?

A:         Create samples Xi drawn from the distribution of X.

Count the number of samples greater than 17 and divide by total number of samples.

## Generating the Samples

Primitive random number generator X ~ U(0,1).

To create Y ~ U(a,b) use

Y = a + ( b – a ) X

In general we can use the cumulative distribution function

1987 – Stochastic Simulation (Ripley) for generating samples for “standard stuff” in textbooks

## Markov Chain Monte Carlo (MCMC) Technique

• Define a suitable Markov Chain whose equilibrium distribution is the desired posterior distribution
• Generate samples from the Markov Chain

## Markov Chain Basics

Example:

This transition probabilities can be written as a matrix

If we write the probability distribution at time t as p(t) then

p(t+1) = p(t)P

For example if the drunk’s walk starts at position 2 we denote

p( t = 0 ) = [0 1 0 0]

p(t) is an evolving probability distribution which is a row vector that sums to one

The equilibrium distribution p(infinity) = p satisfies

pP = p

and is a left eigenvector of P with eigenvalue one.

## Finding the Markov Chain Corresponding to the Posterior Distribution

Metropolis Sampler – Rosenberg, Teller, Teller

Heat Bath ( Gibbs Sampler ) – “rapidly mixing” determines convergence rate

## Metropolis Sampler

We are given that the posterior distribution is of the form f(x)/Z

1.      We have a proposal kernal satisfying K(x,y) = K(y,x)

2.      Calculate f(y)

3.      Accept transition with probability = min {1, f(y)/f(x) }

4.      This eventually converges to the “right thing”