Bayesian Network Fundamentals

Bayesian Network Fundamentals

• A Bayesian network is:

  1. A directed acyclic graph (DAG) G whose nodes represent the random variables X1,X2,...,Xn
  2. For each node Xi a CPD P(Xi|ParG(Xi))

• The BN represents a joint distribution via the chain rule for Bayesian Networks
  

  P(X1,X2,...,Xn)=∏iP(Xi|ParG(Xi))

• BN is a legal distribution:
  

  P(X1,X2,...,Xn)≥0
  
  
  ∑iP(X1,X2,...,Xn)=1

• P factorized over G

  1. Let G be a graph over X1,X2,...,Xn
  2. P factorized over G if
     
     P(X1,X2,...,Xn)=∏iP(Xi|ParG(Xi))

• Reasoning patterns:

  1. Causal reasoning
  2. Evidential reasoning
  3. Intercausal reasoning

• Active Trail

  1. A trail X1−X2−...−Xk is active if it has no v-structures Xi−1→Xi←Xi+1
  2. A trail X1−X2−...−Xk is active given Z if:
     
     • for any v-structure Xi−1→Xi←Xi+1 , we have that Xi or one of its descendants ∈Z
     • no other Xi is in Z