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.
Given an observable random variable X over the probability space
(
X
,
Σ
,
P
θ
)
, determined by a parameter θ ∈ Θ, and a set A of possible actions, a (deterministic) decision rule is a function δ :
X
→ A.
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
θ
, the domain of
θ
may extend over
R
(all real numbers). An associated decision rule for estimating
θ
from some observed data might be, "choose the value of the
θ
, say
θ
^
, that minimizes the sum of squared error between some observed responses and responses predicted from the corresponding covariates given that you chose
θ
^
." Thus, the cost function is the sum of squared error, and one would aim to minimize this cost. Once the cost function is defined,
θ
^
could be chosen, for instance, using some optimization algorithm.
Out of sample prediction in regression and classification models.
