Probabilistic Decision Support System: Influence Diagrams

From DSL

Influence diagrams (Howard & Matheson 1984), also called relevance diagrams, are directed acyclic graphs representing decision problems. The goal of influence diagram modeling is choosing such a decision alternative that will lead to the highest expected gain (utility).

Similarly to Bayesian Networks, influence diagrams are very useful in showing the structure of the domain, i.e., the structure of the decision problem. Influence diagrams contain four types of nodes (Decision, Chance, Deterministic, and Value) and two types of arcs (influences and informational arcs). The following influence diagram models a decision related to investment in a risky venture:

A simple Influence Diagram

Nodes in an influence diagram represent various types of variables:

• Decision nodes, usually drawn as rectangles (such as node Invest above), represent variables that are under control of the decision maker and model the decision alternatives available to the decision maker. Decision nodes include a specification of the available decision options. Node Invest above has two alternatives Invest and DoNotInvest.

• Chance nodes, usually drawn as circles or ovals (such as nodes Forecast and Success above), are random variables and they represent uncertain quantities that are relevant to the decision problem. They are quantified by conditional probability distributions, identical as those presented in the section on Bayesian Networks. In fact, the subset of an influence diagram that consists of only chance nodes is a Bayesian Network, i.e., an influence diagram can be also viewed as Bayesian Network extended by decision and value nodes. Predecessors of chance nodes that are Decision nodes act in exactly the same way as those predecessors that are Chance nodes - they index the conditional probability table of the child node.

• Deterministic nodes, usually drawn as double-circles or double-ovals, represent either constant values or values that are algebraically determined from the states of their parents. In other words, if the values of their parents are known, then the value of a deterministic node is also known with certainty. Deterministic nodes are quantified similarly to Chance nodes. The only difference is that their probability tables contain all zeros or ones (note that there is no uncertainty about the outcome of a deterministic node once all its parents are known).

• Values nodes, usually drawn as diamonds (such as node Gain above), represent utility, i.e., a measure of desirability of the outcomes of the decision process. They are quantified by the utility of each of the possible combinations of outcomes of the parent nodes.

Normally, an arcs in an influence diagram denotes an influence, i.e., the fact that the node at the tail of the arc influences the value (or the probability distribution over the possible values) of the node at the head of the arc. Some arcs in influence diagrams have clearly a causal meaning. In particular, a directed path from a decision node to a chance node means that the decision (i.e., a manipulation of the graph) will impact that chance node in the sense of changing its probability distribution.

Arcs coming into decision nodes have a different meaning. As decision nodes are under decision maker's control, these arcs do not denote influences but rather temporal precedence (in the sense of flow of information). The outcomes of all nodes at the tail of informational arcs will be known before the decision will need to be made. In particular, if there are multiple decision nodes, they need to be all connected by informational arcs. This reflects the fact that the decisions are made in a sequence and the outcome of each decision is known before the next decision is made.