Belief Networks |
We start with a total ordering X_{1},...,X_{n} of the random variables.
Definition. The parents of random variable X_{i}, written pi_{Xi}, are a minimal^{1} set of the predecessors of X_{i} in the total ordering such that the other predecessors of X_{i} are independent of X_{i} given pi_{Xi}. That is pi_{Xi} subset {X_{1},...,X_{i-1}} such that P(X_{i}|X_{i-1}...X_{1}) = P(X_{i}|pi_{Xi}).
A belief network [22] is an acyclic directed graph, where the nodes are random variables^{2}. We use the terms node and random variable interchangeably. There is an arc from each element of pi_{Xi} into X_{i}. Associated with the belief network is a set of probabilities of the form P(X|pi_{X}), the conditional probability of each variable given its parents (this includes the prior probabilities of those variables with no parents).
By the chain rule for conjunctions and the independence assumption:
This factorization of the joint probability distribution is often given as the formal definition of a belief network.
P(X_{1},...,X_{n}) = PROD_{i = 1}^{n} P(X_{i}|X_{i-1}...X_{1}) = PROD_{i = 1}^{n} P(X_{i}|pi_{Xi})
A | B | C | D | P(e|ABCD) |
a | b | c | d | 0.55 |
a | b | c | ~ d | 0.55 |
a | b | ~ c | d | 0.55 |
a | b | ~ c | ~ d | 0.55 |
a | ~ b | c | d | 0.3 |
a | ~ b | c | ~ d | 0.3 |
a | ~ b | ~ c | d | 0.3 |
a | ~ b | ~ c | ~ d | 0.3 |
~ a | b | c | d | 0.08 |
~ a | b | c | ~ d | 0.08 |
~ a | b | ~ c | d | 0.025 |
~ a | b | ~ c | ~ d | 0.5 |
~ a | ~ b | c | d | 0.08 |
~ a | ~ b | c | ~ d | 0.08 |
~ a | ~ b | ~ c | d | 0.85 |
~ a | ~ b | ~ c | ~ d | 0.5 |
Example. Consider the belief network of Figure *. This represents a factorization of the joint probability distribution:
If the variables are binary^{3}, the first term, P(E|ABCD), requires the probability of E for all 16 cases of assignments of values to A,B,C,D. One such table is given in Figure *.
P(A,B,C,D,E,Y,Z) = P(E|ABCD) P(A|YZ) P(B|YZ)P(C|YZ) P(D|YZ) P(Y) P(Z)
Belief Networks |