A Hybrid Algorithm to Compute Marginal and Joint Beliefs in Bayesian Networks and its complexity

1
A Hybrid Algorithm to Compute Marginal and Joint
Beliefs in Bayesian Networks and its complexity

• Mark Bloemeke
• Artificial Intelligence Laboratory
• University of South Carolina

• Marco Valtorta
• Artificial Intelligence Laboratory
• University of South Carolina

Presentation by
Instructor
Sreeja Vallabhan
Marco Valtorta
2
Abstract

• Methods (algorithms) to Update Probability in
  Bayesian Network
• Using a structure (Clique Tree) and perform local
  message based calculation to extract the belief
  in each variable.
• Using Non Serial Dynamic Programming Techniques
  to extract the belief in some desired group of
  variables.

3
Goal

• Present a hybrid algorithm based on Non Serial
  Dynamic Programming Techniques and possessing the
  ability to retrieve the belief in all single
  variables.

4
Symbolic Probabilistic Inference (SPI)

• Consider the Bayesian Network with DAG G (V,
  E)
• and conditional probability tables
• where are the parents of vi in G.
• Total joint probability using Chain Rule of
  Bayesian Network
• (1)
• Using marginalization to retrieve belief in any
  subset of variables V as
• (2)
• SPI is based on these two equations

5
Symbolic Probabilistic Inference (SPI)

• In SPI, to maintain control over the size and
  time complexity of the resulting tables
• Variables are ordered before calculations
• Summations are pushed down into products

6
Symbolic Probabilistic Inference (SPI)

• Consider the following Bayesian Network
• Assuming that each variable has two states,
  P(A,C) require a total of 92 significant
  operations

From equation (1) and (2), the joint probability
of the variable A and C
7
Symbolic Probabilistic Inference (SPI)

• With a single re-ordering of the terms combined
  by equation (1) followed by the distribution of
  the summation from (2)
• This requires only 32 significant operations.

8
Factor Trees

• Two Stage method for deriving the desired joint
  and single beliefs.
• Creation of Factor Tree.
• Passing algorithm on the Factor Tree to retrieve
  desired joint and single beliefs.

9
Factor Trees Algorithm

• Start by Calculating the optimal factoring order
  for the network given the target set of variables
  whose joint is desired.
• Construct a Binary Tree showing the combination
  of initial probability table and conformal table.
• Label edges between table along which variables
  are marginalized with the variables marginalized
  before combination.
• Add an additional head that has an empty label
  above the current root, a conformal table labeled
  with the target set of variables, that has no
  variables.

10
Factor Trees Algorithm