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Inference. Probabilistic Graphical Models. Variable Elimination. Complexity Analysis. Eliminating Z. Reminder: Factor Product. N k =|Val( X k )|. Cost: (m k -1) N k multiplications. Reminder: Factor Marginalization. N k =|Val( X k )|. Cost: ~ N k additions.

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Presentation Transcript
Complexity analysis

Inference

Probabilistic

Graphical

Models

Variable Elimination

Complexity Analysis



Reminder factor product
Reminder: Factor Product

Nk=|Val(Xk)|

Cost: (mk-1)Nk multiplications


Reminder factor marginalization
Reminder: Factor Marginalization

Nk=|Val(Xk)|

Cost: ~Nkadditions


Complexity of variable elimination
Complexity of Variable Elimination

  • Start with m factors

    • m  n for Bayesian networks

    • can be larger for Markov networks

  • At each elimination step generate

  • At most elimination steps

  • Total number of factors: m*


Complexity of variable elimination1
Complexity of Variable Elimination

  • N = max(Nk) = size of the largest factor

  • Product operations: k(mk-1)Nk

  • Sum operations: kNk

  • Total work is linear in N and m*


Complexity of variable elimination2
Complexity of Variable Elimination

  • Total work is linear in N and m

  • Nk=|Val(Xk)|=O(drk) where

    • d = max(|Val(Xi)|)

    • rk = |Xk| = cardinality of the scope of the kth factor


Complexity example
Complexity Example

C

I

D

D

G

S

L

J

H


Complexity and elimination order
Complexity and Elimination Order

C

  • Eliminate: G

I

D

D

G

S

L

J

H


Complexity and elimination order1
Complexity and Elimination Order

A

A

Eliminate A first:

B1

B1

B2

B2

B3

B3

Bk

Bk

C

C

Eliminate Bi‘s first:


Summary
Summary

  • Complexity of variable elimination linear in

    • size of the model (# factors, # variables)

    • size of the largest factor generated

  • Size of factor is exponential in its scope

  • Complexity of algorithm depends heavily on elimination ordering