Rest

• Representing the Full Joint explicitly (e.g., in the form of a table) is

  infeasible for reasonably large worlds X
• Representation the Full Joint explictily is infeasible for reasonable large world X
• Assume that the network is sparsely connected in that each variable has at most $k \leq N$ parents, where k is a fixed constant :

  • Each r ow in the CPD table of a variable X_i requires 1 number
  • If X_i has k parents, the CPD of X_i has 2^k rows
  • all the CPDS together require less then $N *$$2^k$numbers
  • for a fixed k, O(N) --> lineair in N
  • exponential reduction in complexiti from O(2^N) to O(N) if k <= N
• Example

  A) 2x2x3x2x2

  B) 48-1

  C) 1+1+8+3+2

  
• To construct a Bayesian Network, we need to do 3 things :

  1. Decide on the random variables to be used, and their values/domains
  2. Specfiying the structure of the network
  3. Specifiy the CPDs for each node
• CPDs

  Conditional probaility distributions
• Continuous systems and variables solutions:

  1. Discretisation
  2. Work with continuous variables and CPDs
• Discretisation

  Split the domain of a variable into a fixed number of intervals and represent each continuous value by the label of its interval
• Discretisation

  +

  • all iof our reperensentation/reasoning is directly applicable

  -

  • loss of information
  • complexity
• SOLUTION 2: Work with continuous variables and CPDs :

  • Permit continuous random variables X with $Val(X) \subseteq R$
  • Model CPDs as continuous density functions p(X | Pa(X))

  
  

  ! need to define an infinite number of conditional distributions !
• Linear Gaussian Bayesian Network is a BN all of whose variables are

  continuous, and where all of the variables have Linear Gaussian Models.
• Let Y be a continuous variable with continuous parents X1, . . . , Xk .

  We say that Y has a Linear Gaussian Model if there are parameters

  β0, β1, . . . , βk and σ2 such tha

  A) B1x1

  B) o^2

  C) B0 + B.T x

  D) o^2