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" defined below. This concept is analogous to
Pareto efficiency.
Definition
Define
sets ,
and
, where
are the states of nature,
the possible observations, and
the actions that may be taken. An observation
is distributed as
and therefore provides evidence about the state of nature
. A decision rule is a
function , where upon observing
, we choose to take action
.
Also define 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 ...
, which specifies the loss we would incur by taking action
when the true state of nature is
. Usually we will take this action after observing data
, so that the loss will be
. (It is possible though unconventional to recast the following definitions in terms of a
utility function, which is the negative of the loss.)
Define the
risk 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 "cos ...
as the
expectation
Expectation or Expectations may refer to:
Science
* Expectation (epistemic)
* Expected value, in mathematical probability theory
* Expectation value (quantum mechanics)
* Expectation–maximization algorithm, in statistics
Music
* ''Expectation' ...
: