Semantics of Belief Networks

A network represents the joint probability distribution. Every entry of the JPD can be calculated from the network.

• P(X1=x1 ^ ... ^ Xn=xn) = P(x1,...,xn) = Pi(i=1,n) P(xi|Parents(Xi))
• how do we construct the network ?

A network encodes conditional independence statements.

• P(x1,...,xn) = Pi(i=1,n) P(xi|x(i-1),...,x1)
• P(Xi|X(i-1),...,X1) = P(Xi|Parents(Xi)) (1)
  • provided Parents(Xi) are a subset of {x(i-1),...,x1} i.e., order the nodes breadth first
  • those X(i-1),...,X1 not in Parents(Xi) implies conditional independence between the two
• network construction
  • Algorithm:

            choose variables (nodes) Xi
            order the variables
            while variables are left
              pick variable Xi and add node for it
              select Parents(Xi) such that conditional independence satisfied (1)
              define CPT for Xi
    

  • The designer cannot violate the axioms of probability.
  • Order the variables from causes to effects (causal model).
  • We may not need to specify each entry of CPT, e.g., node represents disjunction of parents.

Conditional Independence Relations

• Are X and Y conditionally independent given evidence E?
  • It depends on whether E "blocks" the path from X to Y [Fig15.4, p445].
  • These are needed for inference.