Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

The main steps [8], [12] of the belief–propagation algorithm of Pearl are
outlined below.
1. During initialization, we set all λ and ∏ messages to 1 and set
∏B(A) messages from root to the prior probability [ P (A1)
P(A2),…., P(Am)] T and define the conditional probability matrices.
Then estimate the prior probabilities at all nodes, starting from the
children of the root by taking the product of transpose of the
conditional probability matrix at the link and the prior probability
vector of the parent. Repeat this for all nodes up to the leaves.
2. Generally the variables at the leaves of the tree are instantiated.
Suppose, the variable E= E2 is instantiated. In that case, we set [7]
λE (B) = [ 0 1 0 0 0 0… 0],
where the second element corresponds to instantiation of E = E2.
3. When a node variable is not instantiated, we calculate its λ values
following the formula, outlined in fig. 9.6.
4. The λ and ∏ messages are sent to the parents and the children of the
instantiated node. For the leaf node there is no need to send the ∏
message. Similarly, the root node need not send the λ message.
5. The propagation continues from the leaf to its parent, then from the
parent to the grandparent, until the root is reached. Then down
stream propagation starts from the root to its children, then from the
children to grandchildren of the root and so on until the leaves are
reached. This is called an equilibrium condition, when the λ and ∏
messages do not change, unless instantiated further. The belief value
at the nodes now reflects the belief of the respective nodes for ‘the car
does not start’ ( in our example tree).
6. When we want to fuse the beliefs of more than one evidence, we can
submit the corresponding λ messages at the respective leaves one
after another, and repeat from step 3, otherwise stop.
The resulting beliefs at each node now appear to be the fusion of the joint
effect of two or more observed evidences.
We presented Pearl’s scheme for evidential reasoning for a tree structure only.
However, the belief propagation scheme of Pearl can also be extended