Cognitive Computer Vision Kingsley Sage and Hilary Buxton Prepared under ECVision Specific Action...

Cognitive Computer Vision

Kingsley Sagekhs20@sussex.ac.ukandHilary Buxtonhilaryb@sussex.ac.uk

Prepared under ECVision Specific Action 8-3http://www.ecvision.org

Lecture 3

Graphical models Probabilistic graphical models

– Directed graphs– Unidirected graphs– Notation– Rolling out over time

What are graphical models?

Represent salient relationships graphically e.g.

What are probabilistic graphical models?

A probabilistic graphical model is a type of probabilistic network that has roots in AI, statistics and neural networks

Provides a clean mathematical formalism that makes it possible to understand the relationships between a wide variety of network based approaches to computation

Allows to see different methods as instances of a broader probabilistic framework

What are probabilistic graphical models?

Probabilistic graphical models use graphs to represent and manipulate joint probability distributions

Graphs can be directed – usually referred to as a belief network or Bayesian network

Graphs can be undirected – usually referred to as a Markov Random Field

A basis for algorithms for computation

Joint probability – a reminder

A probability dependent on more than one variable e.g. p(AND|a,b):

a b p(AND|a,b)

0 0 00 1 01 0 01 1 1

a

bDiscrete case of a logic

AND gateA continuous case where

light values are high p(AND|a,b)

Directed graphs

Intuitively, the notion of causality (although this can be a philosophical argument)

A B, so the value of A directly determines the value of B

P(A,B) = P(B|A).P(A)

A

B

Traffic lights model from lecture 2 as a directed graph

STOP

GO

GETREADYTO GO

GETREADY

TO STOP

OBSERVABLE HIDDEN

observable

hidden

Examples of directed graphs

Hidden Markov Models (later in the course) Kalman filters Factor analysis Independent component analysis Mixtures of Gaussians (later in the course) Probabilistic expert systems The list goes on …

Joint probability – conditional independence

N variables are conditionally independent if one value does not depend on the other e.g:

A B

C

)().(),( iff BPAPBAPBA

Here, A and B are conditionallyindependent:

But A and C and B and C are not:

)().().,|()(),().,|()(

),|(),,(

BPAPBACPCPBAPBACPCP

BACPCBAP

Undirected graphs

Intuitively, the notion of correlation (although this can be a philosophical argument)

A B, so the values of A and B are interdependent

Directed graphs can be converted into undirected graphs (but beyond the scope of this course)

A

B

A undirected graph for a computer vision task

Notation

Squares denote discrete nodes

Circles denote continuous valued nodes

Clear denotes hidden node Shaded denotes observed

node

B

A

C

Rolling out over time

Probabilistic graphical model notation is very good at showing how models are propagated in time

Expose the dependencies between the different elements of the graphical structure

Rolling out our traffic light example over 2 time steps …

STOP

GO

GETREADYTO GO

GETREADY

TO STOP

OBSERVABLE HIDDEN

observable

hidden

observable

hidden

t=1 t=2

Remember the concept of the temporal order of a model ?

observable

hidden

observable

hidden

t=1 t=2 In this model, the value of the

hidden nodes (and thus the observable ones) at time t+1 only depends on the previous time step t

So this is a first order temporal model

Remember the concept of the temporal order of a model ?

A second order temporal model

observable

hidden

observable

hidden

t=1 t=2

observable

hidden

observable

hidden

t=3 t=4

…

…

So why are graphical models relevant to Cognitive CV?

Precisely because they allows us to see different methods as instances of a broader probabilistic framework

These methods are the basis for our model of perception guided by expectation

We can put our model of expectation on a solid theoretical foundation

We can develop well-founded methods of learning rather than just being stuck with hand-coded models

Summary

Probabilistic graphical models put the formalisms on a well-founded mathematical basis

We can distinguish directed and undirected graphs

Here we concentrate on directed graphs that we can roll out over time easily

Next time …

A family of graphical models A lot of excellent reference material can be found at:

http://cosco.hiit.fi/Teaching/GraphicalModels/Fall2003/material.html