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A distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400801.png" /> of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400802.png" /> from a family of distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400803.png" /> of an observation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400804.png" />. It was introduced by R.A. Fisher [[#References|[1]]] for numerical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400806.png" /> in the case when the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400807.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400808.png" /> decreases as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400809.png" /> increases in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008010.png" />, considered as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008011.png" /> for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008012.png" />, has the properties of a distribution function (in such a situation one often makes use of a sufficient statistic in the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008013.png" />).
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A fiducial distribution is defined for invariant families of distributions (cf. [[#References|[2]]]–[[#References|[4]]]). Namely, suppose that a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008014.png" /> of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008015.png" /> acts on the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008017.png" />. A family of distributions is called invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008018.png" /> has the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008019.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008020.png" /> has the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008021.png" />. In this case one considers equivariant decision rules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008022.png" /> (i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008023.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008025.png" />) and invariant loss functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008026.png" /> (i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008030.png" />). If the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008032.png" /> is transitive, then the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008033.png" /> has a certain property of homogeneity: For a fixed parameter value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008034.png" /> and an observation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008035.png" /> with the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008036.png" />, the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008037.png" /> runs through the whole family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008038.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008039.png" /> runs through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008040.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008041.png" /> is a set of probability measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008042.png" /> (it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008043.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008045.png" /> are given such that the transformations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008046.png" /> are measurable). Let the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008048.png" /> be given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008050.png" />. The fiducial distribution is described by the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008051.png" /> of probability measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008052.png" /> that minimize the risk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008053.png" /> in the class of equivariant decision rules for every invariant loss function satisfying the following condition of unbiasedness type
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008054.png" /></td> </tr></table>
+
A distribution  $  P _ {x}  ^ {*} $
 +
of the parameter  $  \theta $
 +
from a family of distributions  $  {\mathcal P} = \{ {P _  \theta  } : {\theta \in \Theta } \} $
 +
of an observation  $  x $.
 +
It was introduced by R.A. Fisher [[#References|[1]]] for numerical  $  \theta $
 +
and  $  x $
 +
in the case when the distribution function  $  F ( x \mid  \theta ) $
 +
of  $  x $
 +
decreases as  $  \theta $
 +
increases in such a way that  $  F ^ { * } ( \theta | x) = 1 - F ( x | \theta ) $,
 +
considered as a function of  $  \theta $
 +
for fixed  $  x $,
 +
has the properties of a distribution function (in such a situation one often makes use of a [[sufficient statistic]] in the role of  $  x $).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008055.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008056.png" />, then the family of the fiducial distributions is uniquely distinguished by the requirements that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008057.png" /> is invariant and that the probable and fiducial probabilities are equal, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008058.png" />, for invariant families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008059.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008060.png" /> is called invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008062.png" /> imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008063.png" />).
+
A fiducial distribution is defined for invariant families of distributions (cf. [[#References|[2]]]–[[#References|[4]]]). Namely, suppose that a group  $  G $
 +
of transformations  $  g $
 +
acts on the sets  $  X $
 +
and  $  \Theta $.
 +
A family of distributions is called invariant if  $  gx $
 +
has the distribution  $  P _ {g \theta }  $
 +
when  $  x $
 +
has the distribution  $  P _  \theta  $.
 +
In this case one considers equivariant decision rules  $  \delta :  X \rightarrow D $(
 +
i.e. such that  $  \delta ( gx) = g \delta ( x) $
 +
for all  $  x $
 +
and  $  g $)
 +
and invariant loss functions  $  L _  \theta  ( d) $(
 +
i.e. such that  $  L _ {g \theta }  ( gd) = L _  \theta  ( d) $
 +
for all  $  \theta $,
 +
$  d $
 +
and  $  g $).
 +
If the action of  $  G $
 +
on  $  \Theta $
 +
is transitive, then the family  $  {\mathcal P} $
 +
has a certain property of homogeneity: For a fixed parameter value  $  \theta _ {0} $
 +
and an observation  $  x $
 +
with the distribution  $  P _ {\theta _ {0}  } $,
 +
the distribution of  $  gx $
 +
runs through the whole family  $  {\mathcal P} $
 +
as  $  g $
 +
runs through  $  G $.  
 +
Suppose that  $  D $
 +
is a set of probability measures on  $  \Theta $(
 +
it is assumed that  $  \sigma $-
 +
algebras  $  {\mathcal B} ( \Theta ) $
 +
and  $  {\mathcal B} ( X) $
 +
are given such that the transformations in  $  G $
 +
are measurable). Let the action of  $  G $
 +
on $  D $
 +
be given by  $  ( g \alpha ) ( B) = \alpha ( g  ^ {-} 1 ( B)) $,
 +
$  G \in {\mathcal B} ( \Theta ) $.
 +
The fiducial distribution is described by the family  $  {\mathcal P}  ^ {*} = \{ {P _ {x}  ^ {*} } : {x \in X } \} $
 +
of probability measures on  $  \Theta $
 +
that minimize the risk  $  \int L _  \theta  ( \delta ( x))  dP _  \theta  ( x) $
 +
in the class of equivariant decision rules for every invariant loss function satisfying the following condition of unbiasedness type
 +
 
 +
$$
 +
\int\limits L _  \theta  ( \alpha ) \
 +
d \beta ( \theta )  \geq  \
 +
\int\limits L _  \theta  ( \beta ) \
 +
d \beta ( \theta ).
 +
$$
 +
 
 +
If  $  G $
 +
acts transitively on  $  X $,  
 +
then the family of the fiducial distributions is uniquely distinguished by the requirements that $  {\mathcal P}  ^ {*} = \{ {P _ {x}  ^ {*} } : {x \in X } \} $
 +
is invariant and that the probable and fiducial probabilities are equal, $  P _  \theta  \{ \theta \in S ( x) \} = P _ {x}  ^ {*} \{ \theta \in S ( x) \} $,  
 +
for invariant families $  S ( x) $(
 +
$  S ( x) $
 +
is called invariant if $  \theta \in S ( x) $,  
 +
$  g \in G $
 +
imply that $  g \theta \in S ( gx) $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.A. Fisher,  "Inverse probability"  ''Proc. Cambridge Philos. Soc.'' , '''26'''  (1930)  pp. 528–535</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.A.S. Fraser,  "The fiducial method and invariance"  ''Biometrika'' , '''48'''  (1961)  pp. 261–280</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.P. Klimov,  "On the fiducial approach in statistics"  ''Soviet Math. Dokl.'' , '''11''' :  2  (1970)  pp. 442–444  ''Dokl. Akad. Nauk SSSR'' , '''191''' :  4  (1970)  pp. 763–765</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.P. Klimov,  "Invariant inferences in statistics" , Moscow  (1973)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.A. Fisher,  "Inverse probability"  ''Proc. Cambridge Philos. Soc.'' , '''26'''  (1930)  pp. 528–535</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.A.S. Fraser,  "The fiducial method and invariance"  ''Biometrika'' , '''48'''  (1961)  pp. 261–280</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.P. Klimov,  "On the fiducial approach in statistics"  ''Soviet Math. Dokl.'' , '''11''' :  2  (1970)  pp. 442–444  ''Dokl. Akad. Nauk SSSR'' , '''191''' :  4  (1970)  pp. 763–765</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.P. Klimov,  "Invariant inferences in statistics" , Moscow  (1973)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 10:31, 16 July 2021


A distribution $ P _ {x} ^ {*} $ of the parameter $ \theta $ from a family of distributions $ {\mathcal P} = \{ {P _ \theta } : {\theta \in \Theta } \} $ of an observation $ x $. It was introduced by R.A. Fisher [1] for numerical $ \theta $ and $ x $ in the case when the distribution function $ F ( x \mid \theta ) $ of $ x $ decreases as $ \theta $ increases in such a way that $ F ^ { * } ( \theta | x) = 1 - F ( x | \theta ) $, considered as a function of $ \theta $ for fixed $ x $, has the properties of a distribution function (in such a situation one often makes use of a sufficient statistic in the role of $ x $).

A fiducial distribution is defined for invariant families of distributions (cf. [2][4]). Namely, suppose that a group $ G $ of transformations $ g $ acts on the sets $ X $ and $ \Theta $. A family of distributions is called invariant if $ gx $ has the distribution $ P _ {g \theta } $ when $ x $ has the distribution $ P _ \theta $. In this case one considers equivariant decision rules $ \delta : X \rightarrow D $( i.e. such that $ \delta ( gx) = g \delta ( x) $ for all $ x $ and $ g $) and invariant loss functions $ L _ \theta ( d) $( i.e. such that $ L _ {g \theta } ( gd) = L _ \theta ( d) $ for all $ \theta $, $ d $ and $ g $). If the action of $ G $ on $ \Theta $ is transitive, then the family $ {\mathcal P} $ has a certain property of homogeneity: For a fixed parameter value $ \theta _ {0} $ and an observation $ x $ with the distribution $ P _ {\theta _ {0} } $, the distribution of $ gx $ runs through the whole family $ {\mathcal P} $ as $ g $ runs through $ G $. Suppose that $ D $ is a set of probability measures on $ \Theta $( it is assumed that $ \sigma $- algebras $ {\mathcal B} ( \Theta ) $ and $ {\mathcal B} ( X) $ are given such that the transformations in $ G $ are measurable). Let the action of $ G $ on $ D $ be given by $ ( g \alpha ) ( B) = \alpha ( g ^ {-} 1 ( B)) $, $ G \in {\mathcal B} ( \Theta ) $. The fiducial distribution is described by the family $ {\mathcal P} ^ {*} = \{ {P _ {x} ^ {*} } : {x \in X } \} $ of probability measures on $ \Theta $ that minimize the risk $ \int L _ \theta ( \delta ( x)) dP _ \theta ( x) $ in the class of equivariant decision rules for every invariant loss function satisfying the following condition of unbiasedness type

$$ \int\limits L _ \theta ( \alpha ) \ d \beta ( \theta ) \geq \ \int\limits L _ \theta ( \beta ) \ d \beta ( \theta ). $$

If $ G $ acts transitively on $ X $, then the family of the fiducial distributions is uniquely distinguished by the requirements that $ {\mathcal P} ^ {*} = \{ {P _ {x} ^ {*} } : {x \in X } \} $ is invariant and that the probable and fiducial probabilities are equal, $ P _ \theta \{ \theta \in S ( x) \} = P _ {x} ^ {*} \{ \theta \in S ( x) \} $, for invariant families $ S ( x) $( $ S ( x) $ is called invariant if $ \theta \in S ( x) $, $ g \in G $ imply that $ g \theta \in S ( gx) $).

References

[1] R.A. Fisher, "Inverse probability" Proc. Cambridge Philos. Soc. , 26 (1930) pp. 528–535
[2] D.A.S. Fraser, "The fiducial method and invariance" Biometrika , 48 (1961) pp. 261–280
[3] G.P. Klimov, "On the fiducial approach in statistics" Soviet Math. Dokl. , 11 : 2 (1970) pp. 442–444 Dokl. Akad. Nauk SSSR , 191 : 4 (1970) pp. 763–765
[4] G.P. Klimov, "Invariant inferences in statistics" , Moscow (1973) (In Russian)

Comments

There has been a continued controversy as to what Fisher really meant by fiducial distributions, many authors finding the idea quite meaningless. See, e.g., [a1] for a recent survey.

References

[a1] J.G. Pedersen, "Fiducial inference" Internat. Stat. Rev. , 46 (1978) pp. 147–170
How to Cite This Entry:
Fiducial distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fiducial_distribution&oldid=18272
This article was adapted from an original article by A.D. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article