Markov Random Fields

Markov Random Fields

Presented by:Vladan Radosavljevic

Outline

• IntuitionSimple Example

• TheorySimple Example - Revisited

• Application

• Summary

• References

Intuition• Simple example

• Observation

– noisy image – pixel values are -1 or +1

• Objective

– recover noise free image

Intuition• An idea

• Represent pixels as random variables

• y - observed variable• x - hidden variable

x and y are binary variables (-1 or +1)

• Question: – Is there any relation

among those variables?

Intuition• Building a model

• Values of observed and original pixels should be correlated (small level of noise) - make connections!

• Values of neighboring pixels should be correlated (large homogeneous areas, objects) - make connections!

Final model

Intuition• Why do we need a model?

• y is given, x has to be find• The objective is to find an image x that maximizes p(x|y)• Use model

– penalize connected pairs in the model that have opposite sign as they are not correlated

• Assume distribution

p(x,y) ~ exp(-E)

E =

over all pairs of connected nodes

• If xi and xj have the same sign, the probability will be higher• The same holds for x and y

Intuition• How to find an image x which maximizes probability

p(x|y)?

• Assume x=y

• Take a node xi at time and evaluate E for

xi=+1 and xi=-1

• Set xi to the value that has lowest E (highest probability)

• Iterate through all nodes until convergence

• This method finds local optimum

Intuition

• Result

Theory• Graphical models – a general framework for

representing and manipulating joint distributions defined over sets of random variables• Each variable is associated with a node in a graph• Edges in the graph represent dependencies between

random variables– Directed graphs: represent causative relationships (Bayesian

Networks)– Undirected graphs: represent correlative relationships (Markov

Random Fields)• Representational aspect: efficiently represent complex

independence relations• Computational aspect: efficiently infer information about data

using independence relations

Theory• Recall

• If there are M variables in the model each having K possible states, straight forward inference algorithm will be exponential in the size of the model (KM)

• However, inference algorithms (whether computing distributions, expectations etc.) can use structure in the graph for the purposes of efficient computation

Theory• How to use information from the structure?

• Markov property: If all paths that connect nodes in set A to nodes in set B pass through nodes in set C, then we say that A and B are conditionally independent given C:

p(A,B|C) = p(A|C)p(B|C)• The main idea is to factorize joint probability, then

use sum and product rules for efficient computation

Theory• General factorization

• If two nodes xi and xk are not connected, then they have to be conditionally independent given all other nodes

• There is no link between those two nodes and all other links pass through nodes that are observed

• This can be expressed as

• Therefore, joint distribution must be factorized such that unconnected nodes do not appear in the same factor

• This leads to the concept of clique: a subset of nodes such that all pairs of nodes in the subset are connected

• The factors are defined as functions on the all possible cliques

Theory• Example

• Factorization

where : potential function on the clique C Z : partition function

Theory• Since potential functions have to be positive we can

define them as:

(there is also a theorem that proofs correspondence of this distribution and Markov Random Fields, E is energy function)

• Recall

E =

Theory• Why is this useful?• Inference algorithms can take an advantage of such

representation to significantly increase computational efficiency

• Example, inference on a chain

• To find marginal distribution ~ KN

Theory• If we rearrange the order of summations and

multiplications ~ NK2

Application

Summary

• Advantages• Graphical representation• Computational efficiency

• Disadvantages• Parameter estimation• How to define a model?• Computing probability is sometimes difficult

References

[1] Alexander T. Ihler, Sergey Kirshner, Michael Ghilc, Andrew W. Robertson and Padhraic Smyth “Graphical models for statistical inference and data assimilation”, Physica D: Nonlinear Phenomena, Volume 230, Issues 1-2, June 2007, Pages 72-87