# Decision rule

In decision theory, a decision rule is a function which maps an observation to an appropriate action. Decision rules play an important role in the theory of statistics and economics, and are closely related to the concept of a strategy in game theory.

In order to evaluate the usefulness of a decision rule, it is necessary to have a loss function detailing the outcome of each action under different states.

## Formal definition

Given an observable random variable X over the probability space ${\displaystyle \scriptstyle ({\mathcal {X}},\Sigma ,P_{\theta })}$, determined by a parameter θ ∈ Θ, and a set A of possible actions, a (deterministic) decision rule is a function δ : ${\displaystyle \scriptstyle {\mathcal {X}}}$→ A.

## Examples of decision rules

• An estimator is a decision rule used for estimating a parameter. In this case the set of actions is the parameter space, and a loss function details the cost of the discrepancy between the true value of the parameter and the estimated value. For example, in a linear model with a single scalar parameter ${\displaystyle \theta }$, the domain of ${\displaystyle \theta }$ may extend over ${\displaystyle {\mathcal {R}}}$ (all real numbers). An associated decision rule for estimating ${\displaystyle \theta }$ from some observed data might be, "choose the value of the ${\displaystyle \theta }$, say ${\displaystyle {\hat {\theta }}}$, that minimizes the sum of squared error between some observed responses and responses predicted from the corresponding covariates given that you chose ${\displaystyle {\hat {\theta }}}$." Thus, the cost function is the sum of squared error, and one would aim to minimize this cost. Once the cost function is defined, ${\displaystyle {\hat {\theta }}}$ could be chosen, for instance, using some optimization algorithm.
• Out of sample prediction in regression and classification models.