In
decision theory
Decision theory or the theory of rational choice is a branch of probability theory, probability, economics, and analytic philosophy that uses expected utility and probabilities, probability to model how individuals would behave Rationality, ratio ...
, 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 (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
and
economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interac ...
, and are closely related to the concept of a
strategy
Strategy (from Greek στρατηγία ''stratēgia'', "troop leadership; 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
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...
.
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 "cost ...
detailing the outcome of each action under different states.
Formal definition
Given an observable random variable ''X'' over the
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
, 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
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
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
, the domain of
may extend over
(all real numbers). An associated decision rule for estimating
from some observed data might be, "choose the value of the
, say
, that minimizes the sum of squared error between some observed responses, and responses predicted from the corresponding covariates given that you chose
." Thus, the cost function is the sum of squared error, and one would aim to minimize this cost. Once the cost function is defined,
could be chosen, for instance, using some optimization algorithm.
* Out of sample
prediction
A prediction (Latin ''præ-'', "before," and ''dictum'', "something said") or forecast is a statement about a future event or about future data. Predictions are often, but not always, based upon experience or knowledge of forecasters. There ...
in
regression and
classification
Classification is the activity of assigning objects to some pre-existing classes or categories. This is distinct from the task of establishing the classes themselves (for example through cluster analysis). Examples include diagnostic tests, identif ...
models.
See also
*
Admissible decision rule
*
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 Given a population whose members each belong to one of a number of different sets or classes, a classification rule or classifier is a procedure by which the elements of the population set are each predicted to belong to one of the classes. A perfe ...
*
Scoring rule
In decision theory, a scoring rule
provides evaluation metrics for probabilistic forecasting, probabilistic predictions or forecasts. While "regular" loss functions (such as mean squared error) assign a goodness-of-fit score to a predicted value ...
{{Decision theory
Decision theory