In decision theory

Decision theory

Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...

, 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

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

 and economics

Economics

Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, and are closely related to the concept of a strategy

Strategy (game theory)

In game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise; this fully determines the player's behaviour...

 in game theory

Game theory

Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

.

In order to evaluate the usefulness of a decision rule, it is necessary to have a loss function

Loss function

In statistics and decision theory a loss function is a function that maps an event onto a real number intuitively representing some "cost" associated with the event. Typically it is used for parameter estimation, and the event in question is some function of the difference between estimated and...

 detailing the outcome of each action under different states.

Formal definition

Given an observable random variable X over the probability space

Probability space

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

 , determined by a parameter θ ∈ Θ, and a set A of possible actions, a (deterministic) decision rule is a function δ : → A.

Examples of decision rules

• An estimator
  Estimator
  In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....
  
   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.
• Out of sample prediction
  Prediction
  A prediction or forecast is a statement about the way things will happen in the future, often but not always based on experience or knowledge...
  
   in regression
  Regression analysis
  In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
  
   and classification models.