PGM CH 4.1-4.2 NOTESLAC group, 16/06/2011

So far...

Directed graphical models Bayesian Networks

Useful because both the structure and the parameters provide a natural representation for many types of real-world domains.

This chapter...

Undirected graphical models

Useful in modelling phenomena where we cannot determine the directionality of the interaction between the variables.

Offer a different, simpler perspective on directed models (both independence structure & inference task)

This chapter...

Introduce a framework that allows both directed and undirected edges

Note: some of the results in this chapter require that we restrict attention to distribution over discrete state spaces.

Discrete vs. continuous = boolean or real numbers e.g. 2.1.6

The 4 students example

(The misconception example sec. 3.4.2, ex.3.8)

4 students who get together in pairs to work on their homework for a class. The pairs that meet are shown via the edges (lines) of this undirected graph : A : Alice B : Bobby C : Charles D : Debbie

A

D B

C

The 4 students example

We want to model the following distribution:

1) A is independent of C given B and D2) B is independent of D given A and C

}),{|()2 CADB

}),{|()1 DBCA

The 4 students example

PROBLEM 1:

If we try to model these on a Bayesian network, we will be in trouble:

Any bayesian network I-map of such a distribution will have extraneous edges

At least one of the desired independence statements will not be captured

(cont’d)

The 4 students example

(cont’d) Any bayesian will require from us to

describe the directionality of the influence

Also: Interactions look symmetrical and we

would like to model this somehow, without representing a direction of influence.

The 4 students example

SOLUTION 1:

Undirected graph

= (here) Markov network structure

Nodes (circles) represent variables Edges (lines) represent a notion of direct

probabilistic interaction between the neighbouring variables, not mediated by any other variable in the network.

A

D B

C

The 4 students example

PROBLEM 2: How to parameterise this undirected

graph? CPD (conditional probability

distribution) not useful, as the interaction is not directed

We would like to capture the affinities between the related variables e.g. Alice and Bobby are more likely to agree than disagree

A

D B

C

The 4 students example

SOLUTION 2: Associate A and B with a general

purpose function : factor

The 4 students example

Here we focus only on non-negative factors.

Factor: Let D be a set of random variables. We define a

factor φ to be a function from Val(D) to R. A factor is non-negative if all its entries are non-negative.

Scope:The set of variables D is called the scope of the

factor and is denoted as Scope[φ].

The 4 students example

Let’s calculate the factor of A and B i.e. the fact that Alice and Bob are more likely to agree than disagree:

φ1(A,B) : Val(A,B) to R+

The value associated with a particular assignment a,b denotes the affinity between the two values: the higher the value of φ1(A,B) the more compatible the two values are

The 4 students example

Fig 4.1/a shows one possible compatibility factor for A and B

Not normalised (see partial function later on how to do this)

0: right, 1:wrong/has the misconception

φ1(A,B)

a0 b0 30a0 b1 5a1 b0 1a1 b1 10

0: right, 1:wrong/has the misconception

The 4 students example

φ1(A,B) asserts that: it is more likely that Alice

and Bob agree φ1(a0, b0), φ1(a1, b1) - they are more likely to be either both wrong or both right

If they disagree, Alice is more likely to be right (φ1(a0, b1)) than Bob (φ1(a1, b0))

φ1(A,B)

a0 b0 30a0 b1 5a1 b0 1a1 b1 10

0: right, 1:wrong/has the misconception

The 4 students example

φ3(C,D) asserts that: Charles and Debbie

argue all the time and they will end up disagreeing any way : φ3(c0, d1) and φ3(c1, d0)

φ3(C,D)

c0 d0 1c0 d1 10

0c1 d0 10

0c1 d1 10: right, 1:wrong/has the misconception

The 4 students example

So far: defined the local interactions

between variables/nodes/circles

Next step: Define a global model : need to

combine these interactions = multiply them as with a Bayesian network

The 4 students example

A possible GLOBAL MODEL:

P(a,b,c,d) = φ1(a, b) ∙ φ2(b, c) ∙ φ3(c, d) ∙ φ4(d, a)

PROBLEM:Nothing guarantees that the result is a

normalised distribution (see fig. 4.2 middle column)

The 4 students example

SOLUTIONTake the product of the local factors and normalise it:

P(a,b,c,d) = 1/Z ∙ φ1(a, b) ∙ φ2(b, c) ∙ φ3(c, d) ∙ φ4(d, a)

Where

Z= ∑ φ1(a, b) ∙ φ2(b, c) ∙ φ3(c, d) ∙ φ4(d, a)

Z is a normalising constant known as partition function :

partition as in markov random field in statistical physics;

function , as Z is a function of the parameters [important for machine learning]

The 4 students example

See figure 4.2 for the calculations of the joint distribution

Calculate the partition function of a1,b1,c0,d1

The 4 students example

We can use the partition function/joint probability to answer questions like:

How likely is Bob to have a misconception?

How likely is Bob to have the misconception, given that Charles doesn’t?

The 4 students example

How likely is Bob to have the misconception?

P(b1) ≈ 0.732P(b0) ≈ 0.268

Bob is 26% less ?? likely to have the misconception

The 4 students example

How likely is Bob to have the misconception, given that Charles doesn’t?

P(b1|c0) ≈ 0.06

The 4 students example

Advantages of this approach:

Allows great flexibility in representing interactions between variables. We can change the nature of interaction

between A and B by simply modifying the entries in the factor without caring about normalisation constraints and the interaction of other factors

The 4 students example

Tight connection between factorisation of the distribution and its independence properties:

Factorisation:

),(),()(

:)|(|)3

21 ZYZXXP

asPwritecanweifZYXP

The 4 students example

Using the formula in 3) we can decompose the distribution in several ways e.g.

P(A,B,C,D) = [1/Z ∙ φ1(A, B) ∙ φ2(B, C)] ∙ φ3(C, D) ∙ φ4(A, D)

and infer that

),|(|

),|(|

DBCAP

andCADBP