Causal net, belief network, Bayes

Causal network (belief network)

A causal network, also called a Bayesian belief network, is a directed acyclic graph (DAG) in which each node is a random variable (event or condition) and a directed arc from A to B means "A directly influences / causes B". Every node stores a conditional probability table (CPT), P(node | parents). A node with no parents has a prior probability. The joint probability is P(X1, ..., Xn) = Π P(Xi | Parents(Xi)).

Compared to a full joint table, it needs far fewer probabilities because of the conditional independence between nodes.

Reasoning with Bayes' theorem

Bayes' theorem: P(H | E) = P(E | H) · P(H) / P(E), where P(E) is obtained by summing the joint probability over all possible values of the hidden variables. In a belief network, we compute the required joint probabilities from the CPTs, add them (marginalise), and divide. Reasoning may be causal (cause → effect), diagnostic (effect → cause) or intercausal.

Example

Rain (R) and Sprinkler (S) can cause Wet Grass (W).

   Rain         Sprinkler
  P(R)=0.2       P(S)=0.1
      \           /
       v         v
        Wet Grass

CPT P(W = true | R, S): R,S = (T,T): 0.99; (T,F): 0.90; (F,T): 0.90; (F,F): 0.00.

Query: the grass is wet; what is the probability that it rained, P(R | W)?

Joint probabilities P(R,S,W) = P(R)·P(S)·P(W | R,S):

• R, S: 0.2 × 0.1 × 0.99 = 0.0198
• R, ¬S: 0.2 × 0.9 × 0.90 = 0.1620
• ¬R, S: 0.8 × 0.1 × 0.90 = 0.0720
• ¬R, ¬S: 0.8 × 0.9 × 0.00 = 0

P(R, W) = 0.0198 + 0.1620 = 0.1818

P(W) = 0.1818 + 0.0720 + 0 = 0.2538

P(R | W) = P(R, W) / P(W) = 0.1818 / 0.2538 ≈ 0.716

The prior probability of rain was 0.2, but seeing the wet grass raises it to about 0.72 (diagnostic reasoning). If we also learn that the sprinkler was on, P(R | W, S) = 0.0198 / (0.0198 + 0.0720) ≈ 0.216, which shows that the sprinkler "explains away" the wet grass (intercausal reasoning).