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decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
, 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 Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analy ...
, and are closely related to the concept of a
strategy Strategy (from Greek στρατηγία ''stratēgia'', "art of troop leader; office of general, command, generalship") is a general plan to achieve one or more long-term or overall goals under conditions of uncertainty. In the sense of the " a ...
in game theory. In order to evaluate the usefulness of a decision rule, it is necessary to have a
loss function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "co ...
detailing the outcome of each action under different states.


Formal definition

Given an observable random variable ''X'' over the probability space \scriptstyle (\mathcal,\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→ ''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 \theta, the domain of \theta may extend over \mathcal (all real numbers). An associated decision rule for estimating \theta from some observed data might be, "choose the value of the \theta, say \hat, that minimizes the sum of squared error between some observed responses and responses predicted from the corresponding covariates given that you chose \hat." 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 could be chosen, for instance, using some optimization algorithm. * Out of sample prediction in
regression Regression or regressions may refer to: Science * Marine regression, coastal advance due to falling sea level, the opposite of marine transgression * Regression (medicine), a characteristic of diseases to express lighter symptoms or less extent ( ...
and classification models.


See also

*
Admissible decision rule In statistical decision theory, an admissible decision rule is a rule for making a decision such that there is no other rule that is always "better" than it (or at least sometimes better and never worse), in the precise sense of "better" define ...
*
Bayes estimator In estimation theory and decision theory, a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the posterior expected loss). Equivalently, it maximizes the ...
* Classification rule * Scoring rule {{Unreferenced, date=September 2016 Decision theory