6.3.5. Decision Networks

Decision Network

A decision network (also called an influence diagram) is a graphical model that extends a Bayesian network to include actions (decisions) and preferences (utilities), allowing an agent to reason about what it should do, rather than what is true.

A decision network contains three types of nodes:

1. Chance nodes (ovals) represent random variables, like in Bayesian networks. Denoted as $s$.
2. Decision nodes (rectangles) represent choices the agent can make. Denoted as $a$.
3. Utility nodes (diamonds or rounded rectangles) represent the agent’s preferences or payoff (numerical utility). Denoted by the output of the function $U(s,a)$.

The goal in a decision network is to choose the action that maximizes expected utility. Given evidence $e$, the expected utility of an action $a$ is
$\text{EU}(a\mid e)={\sum }_{x_1,\dots ,x_n}P(x_1,\dots ,x_n\mid e)U(a,x_1,\dots ,x_n),$
where the sum ranges over all relevant chance variables. The optimal action is the one achieving the maximum expected utility (MEU), such that
$\text{MEU}(e)={\max }_a{\sum }_{s}P(s\mid e)U(s,a),$
where $U(s,a)$ is the utility of the utility function evaluated at a specific state–action pair.

In practice, we run inference to compute posteriors and select the action with highest expected utility.

Within decision networks, the Value of Perfect Information (VPI) quantifies how much an agent would benefit, in expectation, from observing an additional variable before making a decision. Let $e$ denote the agent’s current evidence and $E^{\prime }$ a potential new observation. The VPI of observing $E^{\prime }$ given $e$ is defined as
$\text{VPI}(E^{\prime }\mid e)=\text{MEU}(e,E^{\prime })-\text{MEU}(e),$
where $\text{MEU}(e)$ is the maximum expected utility without the new observation, and $\text{MEU}(e,E^{\prime })$ is the expected maximum expected utility after observing $E^{\prime }$, averaged over the possible values that $E^{\prime }$ could take.