Embed Size (px)
DESCRIPTION
What are graphical models? Represent salient relationships graphically e.g.
Citation preview
Cognitive Computer Vision
Kingsley [email protected] [email protected]
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
