Junction Tree Algorithm and ArgMax Junction Tree Algorithm, Study notes of Machine Learning

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Machine Learning

Instructor: Tony Jebara

Topic 18

• The Junction Tree Algorithm
• Collect & Distribute
• Algorithmic Complexity
• ArgMax Junction Tree Algorithm

JTA with many cliques

• Problem: what if we have more than two cliques?

1) Update AB & BC

2) Update BC & CD

• Problem: AB has not heard about CD!

After BC updates, it will be inconsistent for AB

• Need to iterate the pairwise updates many times
• This will eventually converge to consistent marginals
• But, inefficient… can we do better?

AB B BC C^ CD

AB B BC C CD

JTA: Collect & Distribute

• Use tree recursion rather than iterate messages mindlessly!

initialize(DAG){ Pick root

Set all variables as: }

collectEvidence(node) {

for each child of node { update1(node,collectEvidence(child)); }

return(node); }

distributeEvidence(node) {

for each child of node {

update2(child,node); distributeEvidence(child); } }

update1(node w,node v) { }

update2(node w,node v) { }

normalize() { }

ψ C i

= p ( x i | π i ), φ S = 1

p X

( C )

= 1 ∑ C ψ C^ **

ψ C

** ∀ C , p X

( S )

= 1 ∑ (^) S^ φ S^ **

φ S

** ∀ S

φ V ∩ W

= ψ

∑ V ( V ∩ W ) V

, ψ W =

φ V^ * ∩ W φ V ∩ W

ψ W

φ V ∩ W

** = ψ

∑ V \ ( V ∩ W ) V

, ψ W =

φ V^ ** ∩ W φ V^ *^ ∩ W

ψ W

Algorithmic Complexity

• The 5 steps of JTA are all efficient:

1) Moralization

Polynomial in # of nodes

2) Introduce Evidence (fixed or constant)

Polynomial in # of nodes (convert pdf to slices)

3) Triangulate (Tarjan & Yannakakis 1984)

Suboptimal=Polynomial, Optimal=NP

4) Construct Junction Tree (Kruskal)

Polynomial in # of cliques

5) Junction Tree Algorithm (Init,Collect,Distribute,Normalize)

Polynomial (linear) in # of cliques, Exponential in Clique Cardinality

Junction Tree Algorithm

• Convert Directed Graph to Junction Tree

• Initialize separators to 1 (and Z=1) and set clique

tables to the CPTs in the Directed Graph

• Run Collect, Distribute, Normalize
• Get valid marginals from all ψ , φ tables

p ( X ) =

1

Z

ψ X

∏ C (^ C )

φ X

∏ S (^ S )

=

1

1

p x 1 , x

p x 3 | x

p x 4 | x

p x 5 | x

p x 6 | x

p x 7 | x

1 × 1 × 1 × 1 × 1

x

1

x

2

x

2

x

3

x

3

x

4

x

5

x

7

x

3

x

5

x

5

x

6

p ( X ) = p x

p x 2 | x

p x 3 | x

p x 4 | x

p x 5 | x

p x 6 | x

p x 7 | x

ArgMax Junction Tree Algorithm

• We can also use JTA for finding the max not the sum

over the joint to get argmax of marginals & conditionals

• Say have some evidence:
• Most likely (highest p) XF?
• What is most likely state of patient with fever & headache?
• Solution: replace sum with max inside JTA update code
• Final potentials are max marginals:
• Highest value in potential is most likely:

p X F , X

( E )

= p x 1 ,…, x n , x n + 1 ,…, x

( N )

X F

= arg max XF p X F , X

( E )

p F

= max x 2 , x 3 , x 4 , x 5 p x 1 = 1 , x 2 , x 3 , x 4 , x 5 , x 6

( =^1 )

= max x 2 p x 2 | x 1

( =^1 ) p^ x

1

( =^1 )max

x 3 p x 3 | x 1

( =^1 )

max x 4

p ( x 4 | x 2 )max x

5

p ( x 5 | x 3 ) p ( x 6 = 1 | x 2 , x 5 )

ψ

** X

( C )

= max U \ C

p ( X )

X

C

= arg max C ψ

** X

( C )

φ V ∩ W

= max V ( V ∩ W ) ψ V , ψ W =

φ V^ * ∩ W φ V ∩ W

ψ W φ V ∩ W

** = max V \ (^) ( V ∩ W ) ψ V , ψ W =

φ V^ ** ∩ W φ V^ *^ ∩ W

ψ W

ArgMax Junction Tree Algorithm

• Why do I need the ArgMax junction tree algorithm?
• Can’t I just compute marginals using the Sum algorithm

and then find the highest value in each marginal???

• No!! Here’s a counter-example:
• Most likely is x 1 *=C and x 2 *=
• But the sub-marginals p(x 1 ) and p(x 2 ) do not reveal this…
• The marginals would falsely imply that is x 1 *=A and x 2 *=

p x 1 , x

=

x 1

A

x 1

B

x 1

C

x 2 = 0

x 2 = 1

. 14. 05. 27 . 24. 20. 10

⎡

⎣

⎢ ⎢

⎤

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⎥ ⎥

p x

= x 2 =^0

x 2 = 1

. 46 . 54

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

p x

=

A B C

1. 38 0. 25 0. 37

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