Graphical Models and Graphical Models and ApplicationsApplications

CNS/EE148

Instructors: M.Polito, P.Perona, R.McEliece

TA: C. Fanti

Example from Medical Example from Medical DiagnosticsDiagnostics

Visit to Asia

Tuberculosis

Tuberculosisor Cancer

XRay Result Dyspnea

BronchitisLung Cancer

Smoking

Patient Information

Medical Difficulties

Diagnostic Tests

What is a graphical model ?What is a graphical model ?A graphical model is a way of representing probabilistic relationships between random variables.

Variables are represented by nodes:

Conditional (in)dependencies are represented by (missing) edges:

Undirected edges simply give correlations between variables (Markov Random Field or Undirected Graphical model):

Directed edges give causality relationships (Bayesian Network or Directed Graphical Model):

“Graphical models are a marriage between probability theory and graph theory.

They provide a natural tool for dealing with two problems that occur throughout applied mathematics and engineering – uncertainty and complexity –

and in particular they are playing an increasingly important role in the design and analysis of machine learning algorithms.

Fundamental to the idea of a graphical model is the notion of modularity – a complex system is built by combining simpler parts.

Probability theory provides the glue whereby the parts are combined, ensuring that the system as a whole is consistent, and providing ways to interface models to data.

The graph theoretic side of graphical models provides both an intuitively appealing interface by which humans can model highly-interacting sets of variables as well as a data structure that lends itself naturally to the design of efficient general-purpose algorithms.

Many of the classical multivariate probabilistic systems studied in fields such as statistics, systems engineering, information theory, pattern recognition and statistical mechanics are special cases of the general graphical model formalism -- examples include mixture models, factor analysis, hidden Markov models, Kalman filters and Ising models.

The graphical model framework provides a way to view all of these systems as instances of a common underlying formalism.

This view has many advantages -- in particular, specialized techniques that have been developed in one field can be

transferred between research communities and exploited more widely.

Moreover, the graphical model formalism provides a natural framework for the design of new systems.“

--- Michael Jordan, 1998.

(Picture by Zoubin Ghahramani and Sam Roweis)

We already know many graphical models:

Plan for the class Plan for the class Introduction to Graphical Models (Polito)

Basics on graphical models and statistics

Learning from data

Exact inference

Approximate inference

Applications to Vision (Perona)

Applications to Coding Theory (McEliece)

Belief Propagation and Spin Glasses

Plan for the class Plan for the class Introduction to Graphical Models (Polito)

Basics on graphical models and statistics

Learning from data

Exact inference

Approximate inference

Applications to Vision (Perona)

Applications to Coding Theory (McEliece)

Belief Propagation and Spin Glasses

Basics on graphical models and statistics

• Basics of graph theory.

• Families of probability distributions associated to directed and undirected graphs.

•Markov properties and conditional independence.

•Statistical concepts as building blocks for graphical models.

•Density estimation, classification and regression.

Basics on graphical models and statistics

Graphs and Families of Probability Distributions

There is a family of probability distributions that can be represented with this graph. X1

X4

X2

X3

X5

X6

X7

1) Every P.D. presenting (at least) the conditional independencies that can be derived from the graph belongs to that family.2) Every P.D. that can be factorized as p(x1,…,x7)=p(x4|x1,x2) p(x7|x4) p(x5|x4) p(x6|x5,x2) p(x3|x2) belongs to that family.

Basics on graphical models and statistics

Building blocks for graphical models

X

Y

p(X)= ??? p(Y|X)= ???? Bayesian approach: every unknown quantity (including parameters) is treated as a random variable.

X

Density estimation

Regression

Classification

Parametric and nonparametric methods

Linear, conditional mixture, nonparametric

Generative and discriminative approach

Q

X

Q

X

X Y

XX

Plan for the class Plan for the class Introduction to Graphical Models (Polito)

Basics on graphical models and statistics

Learning from data

Exact inference

Approximate inference

Applications to Vision (Perona)

Applications to Coding Theory (McEliece)

Belief Propagation and Spin Glasses

Learning from data

• Model structure and parameters estimation.

•Complete observations and latent variables.

•MAP and ML estimation.

•The EM algorithm.

•Model selection.

Structure Observability Method

Known Full ML or MAP estimation

Known Partial EM algorithm

Unknown Full Model selection or model averaging

Unknown Partial EM + model sel. or model aver.

Plan for the class Plan for the class Introduction to Graphical Models (Polito)

Basics on graphical models and statistics

Learning from data

Exact inference

Approximate inference

Applications to Vision (Perona)

Applications to Coding Theory (McEliece)

Belief Propagation and Spin Glasses

Exact inference

• The junction tree and related algorithms.

• Belief propagation and belief revision.

• The generalized distributive law.

• Hidden Markov Models and Kalmann Filtering with graphical models.

Exact inferenceConditional independencies

Given a probability distribution p(X,Y,Z,W),

How can we decide if the groups of variables X and Y

are “conditionally independent” from each other

once the value of the variables Z is assigned ?

With graphical models, we can implement an algorithm

reducing this global problem to a series of local problems

(see Matlab demo of the Bayes-Ball algorithm)

Exact inference Variable elimination and Distributive Law

X1

X4

X2

X3

X5

X7

X6

X5

X4

X3

p(x1,…,x6,x7)=p(x4|x1,x2)p(x6|x4)p(x5|x4) p(x3|x2)p(x7|x2,x5)

Marginalize over x7:

p(x1,…,x6)=x7 [p(x4|x1,x2)p(x6|x4)p(x5|x4)

p(x3|x2)p(x7|x2,x5)]

Applying a “distributive law”:

p(x1,…,x6)=p(x4|x1,x2)p(x6|x4)p(x5|x4)

p(x3|x2) x7 [ p(x7|x2,x5)]

The language of graphical models allows a general formalization of this method.

Exact inferenceJunction graph and message passing

Group random variables which are “fully connected”.

Connect group-nodes with common members: the “junction graph”.

Every node only needs to “communicate” with its neighbors.

If the junction graph is a tree, there is a “message passing” protocol which allows exact inference.

Plan for the class Plan for the class Introduction to Graphical Models (Polito)

Basics on graphical models and statistics

Learning from data

Exact inference

Approximate inference

Applications to Vision (Perona)

Applications to Coding Theory (McEliece)

Belief Propagation and Spin Glasses

Approximate inferenceApproximate inference

•Kullback-Leibler divergence and entropy.

•Variational methods.

•Monte Carlo Methods.

• Loopy junction graphs and loopy belief propagation.

• Performance of loopy belief propagation.

•Bethe approximation of free energy and belief propagation.

Approximate inferenceApproximate inferenceKullback-Leibler divergence and GMKullback-Leibler divergence and GM

The graphical model associated to a P.D. p(x) is too complicated. AND NOW ?!?!?

We choose to approximate p(x) with q(x), obtained by making assumptions on the junction graph corresponding to the graphical model.

Example: eliminate loops, bound the number of nodes linked to each node.

A good criterion for choosing q: mimimize the cross-entropy, or Kullback-Leibler divergence:

dxxq

xpxpqpD

)(

)(log)()||(

Approximate inferenceApproximate inferenceVariational methodsVariational methods

Example: the QMR-DT database

)exp(1)|1(

)exp()|0(

)(

0

)(

0

ij

ijiji

ij

ijiji

adadfp

adadfp

By using the inequality:)(1 Hxx ee

We get the approximation:

)(

0 )exp())(exp()|1(ij

dijiiiii

jaHadfp

The node fi is “unlinked”:

Diseases: d1,d2,d3

Symptoms: f1,f2,f3,f4

p(f|d)=p(fi|d) p(di)

Approximate inferenceApproximate inferenceLoopy belief propagationLoopy belief propagation

When the junction graph is not a tree, inference is not exact: roughly speaking, a message might pass through a node more often, causing trouble.

However, in certain cases, an iterated application of a message passing algorithm converges to a good candidate for the exact solution.