Philosophy:Decision rule

From HandWiki

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 [math]\displaystyle{ \scriptstyle (\mathcal{X},\Sigma, P_\theta) }[/math], determined by a parameter θ ∈ Θ, and a set A of possible actions, a (deterministic) decision rule is a function δ : [math]\displaystyle{ \scriptstyle\mathcal{X} }[/math]→ 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 [math]\displaystyle{ \theta }[/math], the domain of [math]\displaystyle{ \theta }[/math] may extend over [math]\displaystyle{ \mathcal{R} }[/math] (all real numbers). An associated decision rule for estimating [math]\displaystyle{ \theta }[/math] from some observed data might be, "choose the value of the [math]\displaystyle{ \theta }[/math], say [math]\displaystyle{ \hat{\theta} }[/math], that minimizes the sum of squared error between some observed responses and responses predicted from the corresponding covariates given that you chose [math]\displaystyle{ \hat{\theta} }[/math]." Thus, the cost function is the sum of squared error, and one would aim to minimize this cost. Once the cost function is defined, [math]\displaystyle{ \hat{\theta} }[/math] could be chosen, for instance, using some optimization algorithm.
  • Out of sample prediction in regression and classification models.

See also