Removal

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Removal is one of three elementary operations in evaluating well-formed (partial) relevance diagrams. Node A can be removed into chance node B when:

a) A and B have an identical set of direct predecessors (except for A's arrow into B);

b) B is the only direct successor of A.

Mathematically, removal corresponds to different operations when A is a chance node and when A is a decision node.

For general well-formed (partial) relevance diagrams, removal is only defined when A is a chance node—removing it into B corresponds to integrating the joint probability distribution of A and B with respect to A and replacing B’s conditional probability distribution by its resulting marginal probability distribution.

However, when the diagram is a well-formed (partial) decision relevance diagram and B is its value node, removal is defined both when A is a chance node and when A is a decision node.

When A is a chance node, in addition to integrating A’s associated probability distribution into B's, removing A involves calculating the decision-maker’s expected utility with respect to A.
When A is a decision node, removing A corresponds to maximising the decision-maker’s expected utility with respect to A.

Removing A deletes it from the evaluated diagram. The resulting relevance diagram is decision-equivalent to the original, but it no longer represents the same joint probability distribution over its uncertain variables.