Decision rule

From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about decision theory. For the use in computer science, see Decision tree#Decision rules.

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[edit]

Given an observable random variable X over the probability space  \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 : \scriptstyle\mathcal{X}→ A.

Examples of decision rules[edit]

  • 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 \theta, the domain of \theta may extend over \mathcal{R} (all real numbers). An associated decision rule for estimating \theta from some observed data might be, "choose the value of the \theta, say \hat{\theta}, that minimizes the sum of squared error between some observed responses and responses predicted from the corresponding covariates given that you chose \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, \hat{\theta} could be chosen, for instance, using some optimization algorithm.
  • Out of sample prediction in regression and classification models.

See also[edit]