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Dominating decision rule
Encyclopedia
In decision theory
, a decision rule is said to dominate another if the performance of the former is sometimes better, and never worse, than that of the latter.
Formally, let
and
be two decision rules
, and let
be the risk
of rule
for parameter
. The decision rule
is said to dominate the rule
if
for all
, and the inequality is strict for some
.
This defines a partial order on decision rules; the maximal element
s with respect to this order are called admissible decision rule
s.
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 said to dominate another if the performance of the former is sometimes better, and never worse, than that of the latter.
Formally, let
![](http://image.absoluteastronomy.com/images/formulas/6/2/2627489-1.gif)
![](http://image.absoluteastronomy.com/images/formulas/6/2/2627489-2.gif)
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...
, and let
![](http://image.absoluteastronomy.com/images/formulas/6/2/2627489-3.gif)
Risk function
In decision theory and estimation theory, the risk function R of a decision rule, δ, is the expected value of a loss function L:...
of rule
![](http://image.absoluteastronomy.com/images/formulas/6/2/2627489-4.gif)
![](http://image.absoluteastronomy.com/images/formulas/6/2/2627489-5.gif)
![](http://image.absoluteastronomy.com/images/formulas/6/2/2627489-6.gif)
![](http://image.absoluteastronomy.com/images/formulas/6/2/2627489-7.gif)
![](http://image.absoluteastronomy.com/images/formulas/6/2/2627489-8.gif)
![](http://image.absoluteastronomy.com/images/formulas/6/2/2627489-9.gif)
![](http://image.absoluteastronomy.com/images/formulas/6/2/2627489-10.gif)
This defines a partial order on decision rules; the maximal element
Maximal element
In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually...
s with respect to this order are called admissible decision rule
Admissible decision rule
In statistical decision theory, an admissible decision rule is a rule for making a decision such that there isn't any other rule that is always "better" than it, in a specific sense defined below....
s.