User:Maximilian Janisch/Sandbox
to statistical problems
An approach based on the assumption that to any parameter in a statistical problem there can be assigned a definite probability distribution. Any general statistical decision problem is determined by the following elements: by a space of (potential) samples
, by a space
of values of the unknown parameter
, by a family of probability distributions
on
, by a space of decisions
and by a function
, which characterizes the losses caused by accepting the decision
when the true value of the parameter is
. The objective of decision making is to find in a certain sense an optimal rule (decision function)
, assigning to each result of an observation
the decision
. In the Bayesian approach, when it is assumed that the unknown parameter
is a random variable with a given (a priori) distribution
on
the best decision function (Bayesian decision function)
is defined as the function for which the minimum expected loss
, where
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and
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is attained. Thus,
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In searching for the Bayesian decision function , the following remark is useful. Let
,
, where
and
are certain
-finite measures. One then finds, assuming that the order of integration may be changed,
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It is seen from the above that for a given is that value of
for which
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is attained, or, what is equivalent, for which
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is attained, where
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But, according to the Bayes formula
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Thus, for a given ,
is that value of
for which the conditional average loss
attains a minimum.
Example. (The Bayesian approach applied to the case of distinguishing between two simple hypotheses.) Let ,
,
,
;
,
,
. If the solution
is identified with the acceptance of the hypothesis
:
, it is natural to assume that
,
. Then
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implies that is attained for the function
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The advantage of the Bayesian approach consists in the fact that, unlike the losses , the expected losses
are numbers which are dependent on the unknown parameter
, and, consequently, it is known that solutions
for which
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and which are, if not optimal, at least -optimal
, are certain to exist. The disadvantage of the Bayesian approach is the necessity of postulating both the existence of an a priori distribution of the unknown parameter and its precise form (the latter disadvantage may be overcome to a certain extent by adopting an empirical Bayesian approach, cf. Bayesian approach, empirical).
References
[1] | A. Wald, "Statistical decision functions" , Wiley (1950) |
[2] | M.H. de Groot, "Optimal statistical decisions" , McGraw-Hill (1970) |
Maximilian Janisch/Sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximilian_Janisch/Sandbox&oldid=43845