Английская Википедия:Decision rule

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Шаблон:About 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> \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>\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>\theta</math>, the domain of <math>\theta</math> may extend over <math>\mathcal{R}</math> (all real numbers). An associated decision rule for estimating <math>\theta</math> from some observed data might be, "choose the value of the <math>\theta</math>, say <math>\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>\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>\hat{\theta}</math> could be chosen, for instance, using some optimization algorithm.
• Out of sample prediction in regression and classification models.

See also

• Admissible decision rule
• Bayes estimator
• Classification rule
• Scoring rule