Semantics of BNs

• X,Y,Z in X

  X and Y are conditionally independent given Z $(X \bot Y | Z)$ if :

  • forall avlues x in Val(X) (same for ,y,z) : $P(x|y,z) =$$P(x|z)$
  • equivalent to : $P(X,Y|Z) =$$P(X|Z) P(Y|Z)$
• $(L \bot D, I, S)$


  Is L completely independant of the other variables
• L is conditionally independent of D,I,S given G, but not marginally independant

  SO $(L \bot D,I,S | G) but (L \not \bot D,I,S)$

  
• SAT is conditionally independant of D,G,L given I

  
• $(G \not \bot L | I,D)$


  G is not independant of L event if we know I and D, Knowledge of L helps us to better guess G even if we know I and D
• G is conditionally independant of S, given I and D

  
  

  D and I are independant

  
• Given the values of its parents, a variable X is independent from all other

  variables in the network that are not its children and, more generally, its

  descendants.
• information about X’s descendants can change our belief about X

  (via an evidential reasoning process)
• X’s parents “shield” X from causal influences further up in the network
• In a Bayesian Network, every variable is conditionally independant of all its non-descendants, given tis parents
• A Bayesian Network Structure G is a directly acyclic graph whose nodes represent a set of random variables X = {X1, ..., Xn} Given a structure G, the following conditional independencies hold :

  $(X_i \bot NonDesc(X_i) | Pa(X_i)) , \forall X_i \in X$
• In a Bayesian Network, every variable is condtionally independant of all its non-descendants, given its parents
• If the network graph correctly represents the conditional independencies in

  the joint distribution P over X , then this full joint distribution P can be

  reconstructed from products of the local CPDs
• $P(X_1, ... , X_n) =$$\Pi_{i=1}^{n} P(X_i | X_1, ... X_{i-1}) =$$\Pi_{i=1}^{n} P(X_i | Pa(X_i))$

  • The CPDS jointly define a full probability distribution over the variable space X.
  • This full joint distribution can be represented as a product of the CPDs
• A Bayesian Network B is a BN structure graph G where each node X_i is assocaited with a set of CPDs $P(X_i | Pa(X_i)$

  
  

  • P factorises over network structure G
  • A BN is a factorsied representation of the joint distribution over X
  • Each CPD table is called a factor
• Reconstructing the Joint Distributino

  A) D

  B) I

  C) G | D, I

  D) S|I

  E) L|G

  
• THe Full Joint Distribution $P(X_1, ... , X_N)$over X

  • has $2^N$ entries
  • $2^N -1$ independent parameters