# Philosophy: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 $\displaystyle{ (\mathcal{X},\Sigma, P_\theta) }$, 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.