Difference between revisions of "User:Maximilian Janisch/Sandbox"
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− | + | {{TEX|partial}}iv class="Vorlage_Achtung" style="border: 0.18em solid #FF6666; border-left:1em solid #FF6666; margin:0.5em 0em; overflow:hidden; padding:0.5em; text-align: left;"> | |
This page is a copy of the article [[Bayesian approach]] in order to test [[User:Maximilian_Janisch/latexlist|automatic LaTeXification]]. This article is not my work. | This page is a copy of the article [[Bayesian approach]] in order to test [[User:Maximilian_Janisch/latexlist|automatic LaTeXification]]. This article is not my work. | ||
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''to statistical problems'' | ''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153901.png" /> of (potential) samples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153902.png" />, by a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153903.png" /> of values of the unknown parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153904.png" />, by a family of probability distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153905.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153906.png" />, by a space of decisions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153907.png" /> and by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153908.png" />, which characterizes the losses caused by accepting the decision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153909.png" /> when the true value of the parameter is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539010.png" />. The objective of decision making is to find in a certain sense an optimal rule (decision function) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539011.png" />, assigning to each result of an observation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539012.png" /> the decision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539013.png" />. In the Bayesian approach, when it is assumed that the unknown parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539014.png" /> is a random variable with a given (a priori) distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539016.png" /> the best decision function ([[Bayesian decision function|Bayesian decision function]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539017.png" /> is defined as the function for which the minimum expected loss <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539018.png" />, where | + | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153901.png"/> of (potential) samples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153902.png"/>, by a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153903.png"/> of values of the unknown parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153904.png"/>, by a family of probability distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153905.png"/> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153906.png"/>, by a space of decisions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153907.png"/> and by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153908.png"/>, which characterizes the losses caused by accepting the decision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b0153909.png"/> when the true value of the parameter is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539010.png"/>. The objective of decision making is to find in a certain sense an optimal rule (decision function) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539011.png"/>, assigning to each result of an observation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539012.png"/> the decision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539013.png"/>. In the Bayesian approach, when it is assumed that the unknown parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539014.png"/> is a random variable with a given (a priori) distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539015.png"/> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539016.png"/> the best decision function ([[Bayesian decision function|Bayesian decision function]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539017.png"/> is defined as the function for which the minimum expected loss <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539018.png"/>, where |
− | + | \begin{equation} \rho ( \pi , \delta ) = \int _ { \Theta } \rho ( \theta , \delta ) \pi ( d \theta ) \end{equation} | |
and | and | ||
− | <table class="eq" style="width:100%;"> <tr><td | + | <table class="eq" style="width:100%;"> <tbody><tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539020.png"/></td> </tr></tbody></table> |
is attained. Thus, | is attained. Thus, | ||
− | <table class="eq" style="width:100%;"> <tr><td | + | <table class="eq" style="width:100%;"> <tbody><tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539021.png"/></td> </tr></tbody></table> |
− | In searching for the Bayesian decision function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539022.png" />, the following remark is useful. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539026.png" /> are certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539027.png" />-finite measures. One then finds, assuming that the order of integration may be changed, | + | In searching for the Bayesian decision function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539022.png"/>, the following remark is useful. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539023.png"/>, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539024.png"/>, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539025.png"/> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539026.png"/> are certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539027.png"/>-finite measures. One then finds, assuming that the order of integration may be changed, |
− | <table class="eq" style="width:100%;"> <tr><td | + | <table class="eq" style="width:100%;"> <tbody><tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539028.png"/></td> </tr></tbody></table> |
− | + | \begin{equation} = \int \int _ { \Theta } L ( \theta , \delta ( x ) ) p ( x | \theta ) \pi ( \theta ) d \mu ( x ) d \nu ( \theta ) = \end{equation} | |
− | + | \begin{equation} = \int _ { X } d \mu ( x ) [ \int _ { \Theta } L ( \theta , \delta ( x ) ) p ( x | \theta ) \pi ( \theta ) d \nu ( \theta ) ] \end{equation} | |
− | It is seen from the above that for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539031.png" /> is that value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539032.png" /> for which | + | It is seen from the above that for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539031.png"/> is that value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539032.png"/> for which |
− | <table class="eq" style="width:100%;"> <tr><td | + | <table class="eq" style="width:100%;"> <tbody><tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539033.png"/></td> </tr></tbody></table> |
is attained, or, what is equivalent, for which | is attained, or, what is equivalent, for which | ||
− | <table class="eq" style="width:100%;"> <tr><td | + | <table class="eq" style="width:100%;"> <tbody><tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539034.png"/></td> </tr></tbody></table> |
is attained, where | is attained, where | ||
− | + | \begin{equation} p ( x ) = \int _ { \Theta } p ( x | \theta ) \pi ( \theta ) d \nu ( \theta ) \end{equation} | |
But, according to the [[Bayes formula|Bayes formula]] | But, according to the [[Bayes formula|Bayes formula]] | ||
− | + | \begin{equation} \int _ { \Theta } L ( \theta , d ) \frac { p ( x | \theta ) \pi ( \theta ) } { p ( x ) } d \nu ( \theta ) = E [ L ( \theta , d ) | x ] \end{equation} | |
− | Thus, for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539038.png" /> is that value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539039.png" /> for which the conditional average loss <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539040.png" /> attains a minimum. | + | Thus, for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539037.png"/>, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539038.png"/> is that value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539039.png"/> for which the conditional average loss <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539040.png"/> attains a minimum. |
− | Example. (The Bayesian approach applied to the case of distinguishing between two simple hypotheses.) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539044.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539047.png" />. If the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539048.png" /> is identified with the acceptance of the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539049.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539050.png" />, it is natural to assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539052.png" />. Then | + | Example. (The Bayesian approach applied to the case of distinguishing between two simple hypotheses.) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539041.png"/>, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539042.png"/>, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539043.png"/>, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539044.png"/>; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539045.png"/>, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539046.png"/>, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539047.png"/>. If the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539048.png"/> is identified with the acceptance of the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539049.png"/>: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539050.png"/>, it is natural to assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539051.png"/>, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539052.png"/>. Then |
− | + | \begin{equation} \rho ( \pi , \delta ) = \int _ { X } [ \pi _ { 1 } p ( x | \theta _ { 1 } ) L ( \theta _ { 1 } , \delta ( x ) ) + \end{equation} | |
− | <table class="eq" style="width:100%;"> <tr><td | + | <table class="eq" style="width:100%;"> <tbody><tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539054.png"/></td> </tr></tbody></table> |
− | implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539055.png" /> is attained for the function | + | implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539055.png"/> is attained for the function |
− | + | \begin{equation} \delta ^ { * } ( x ) = \left\{ \begin{array} { l l } { d _ { 1 } , } & { \text { if } \frac { p ( x | \theta _ { 2 } ) } { p ( x | \theta _ { 1 } ) } \leq \frac { \pi _ { 1 } } { \pi _ { 2 } } \frac { L _ { 12 } - L _ { 11 } } { L _ { 21 } - L _ { 22 } } } \\ { d _ { 2 } , } & { \text { if } \frac { p ( x | \theta _ { 2 } ) } { p ( x | \theta _ { 1 } ) } \geq \frac { \pi _ { 1 } } { \pi _ { 2 } } \frac { L _ { 12 } - L _ { 11 } } { L _ { 21 } - L _ { 22 } } } \end{array} \right. \end{equation} | |
− | The advantage of the Bayesian approach consists in the fact that, unlike the losses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539057.png" />, the expected losses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539058.png" /> are numbers which are dependent on the unknown parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539059.png" />, and, consequently, it is known that solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539060.png" /> for which | + | The advantage of the Bayesian approach consists in the fact that, unlike the losses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539057.png"/>, the expected losses <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539058.png"/> are numbers which are dependent on the unknown parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539059.png"/>, and, consequently, it is known that solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539060.png"/> for which |
− | + | \begin{equation} \rho ( \pi , \delta _ { \epsilon } ^ { * } ) \leq \operatorname { inf } _ { \delta } \rho ( \pi , \delta ) + \epsilon \end{equation} | |
− | and which are, if not optimal, at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539062.png" />-optimal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539063.png" />, 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|Bayesian approach, empirical]]). | + | and which are, if not optimal, at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539062.png"/>-optimal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015390/b01539063.png"/>, 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|Bayesian approach, empirical]]). |
====References==== | ====References==== | ||
− | <table>< | + | <table><tbody><tr><td valign="top">[1]</td> <td valign="top"> A. Wald, "Statistical decision functions" , Wiley (1950)</td></tr><tr><td valign="top">[2]</td> <td valign="top"> M.H. de Groot, "Optimal statistical decisions" , McGraw-Hill (1970)</td></tr></tbody></table |
Revision as of 12:25, 1 September 2019
iv class="Vorlage_Achtung" style="border: 0.18em solid #FF6666; border-left:1em solid #FF6666; margin:0.5em 0em; overflow:hidden; padding:0.5em; text-align: left;"> This page is a copy of the article Bayesian approach in order to test automatic LaTeXification. This article is not my work.
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
\begin{equation} \rho ( \pi , \delta ) = \int _ { \Theta } \rho ( \theta , \delta ) \pi ( d \theta ) \end{equation}
and
<tbody></tbody>![]() |
is attained. Thus,
<tbody></tbody>![]() |
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,
![]() |
\begin{equation} = \int \int _ { \Theta } L ( \theta , \delta ( x ) ) p ( x | \theta ) \pi ( \theta ) d \mu ( x ) d \nu ( \theta ) = \end{equation}
\begin{equation} = \int _ { X } d \mu ( x ) [ \int _ { \Theta } L ( \theta , \delta ( x ) ) p ( x | \theta ) \pi ( \theta ) d \nu ( \theta ) ] \end{equation}
It is seen from the above that for a given is that value of
for which
![]() |
is attained, or, what is equivalent, for which
<tbody></tbody>![]() |
is attained, where
\begin{equation} p ( x ) = \int _ { \Theta } p ( x | \theta ) \pi ( \theta ) d \nu ( \theta ) \end{equation}
But, according to the Bayes formula
\begin{equation} \int _ { \Theta } L ( \theta , d ) \frac { p ( x | \theta ) \pi ( \theta ) } { p ( x ) } d \nu ( \theta ) = E [ L ( \theta , d ) | x ] \end{equation}
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
\begin{equation} \rho ( \pi , \delta ) = \int _ { X } [ \pi _ { 1 } p ( x | \theta _ { 1 } ) L ( \theta _ { 1 } , \delta ( x ) ) + \end{equation}
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implies that is attained for the function
\begin{equation} \delta ^ { * } ( x ) = \left\{ \begin{array} { l l } { d _ { 1 } , } & { \text { if } \frac { p ( x | \theta _ { 2 } ) } { p ( x | \theta _ { 1 } ) } \leq \frac { \pi _ { 1 } } { \pi _ { 2 } } \frac { L _ { 12 } - L _ { 11 } } { L _ { 21 } - L _ { 22 } } } \\ { d _ { 2 } , } & { \text { if } \frac { p ( x | \theta _ { 2 } ) } { p ( x | \theta _ { 1 } ) } \geq \frac { \pi _ { 1 } } { \pi _ { 2 } } \frac { L _ { 12 } - L _ { 11 } } { L _ { 21 } - L _ { 22 } } } \end{array} \right. \end{equation}
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
\begin{equation} \rho ( \pi , \delta _ { \epsilon } ^ { * } ) \leq \operatorname { inf } _ { \delta } \rho ( \pi , \delta ) + \epsilon \end{equation}
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
<tbody></tbody></table[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