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The concept of a yoke, introduced in [[#References|[a3]]], is of great importance in relation to geometric, i.e. parametrization invariant, calculations on statistical models (cf. also [[Differential geometry in statistical inference|Differential geometry in statistical inference]]; [[Statistical manifold|Statistical manifold]]). A yoke on a model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y1100201.png" /> induces a metric and families of connections, derivative strings and tensors on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y1100202.png" /> in terms of which geometric properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y1100203.png" /> may be formulated, see [[#References|[a5]]]. Differences and similarities between the expected and observed geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y1100204.png" /> may be discussed using yokes, see [[#References|[a5]]]. Furthermore, invariant Taylor expansions of functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y1100205.png" /> are obtainable via yokes. Finally, a relationship between yokes and symplectic forms has been established in [[#References|[a4]]].
+
{{TEX|done}}
  
In order to define a yoke, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y1100206.png" /> be a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y1100207.png" />-dimensional [[Manifold|manifold]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y1100208.png" /> and, correspondingly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y1100209.png" /> denote local coordinates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002011.png" />, respectively. Arbitrary components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002012.png" /> will be denoted by the letters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002013.png" />. For two sets of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002015.png" /> and a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002016.png" />, the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002017.png" /> is used for the values of the function
+
The concept of a yoke, introduced in [[#References|[a3]]], is of great importance in relation to geometric, i.e. parametrization invariant, calculations on statistical models (cf. also [[Differential geometry in statistical inference|Differential geometry in statistical inference]]; [[Statistical manifold|Statistical manifold]]). A yoke on a model  $  M $
 +
induces a metric and families of connections, derivative strings and tensors on  $  M $
 +
in terms of which geometric properties of $  M $
 +
may be formulated, see [[#References|[a5]]]. Differences and similarities between the expected and observed geometry of  $  M $
 +
may be discussed using yokes, see [[#References|[a5]]]. Furthermore, invariant Taylor expansions of functions defined on  $  M $
 +
are obtainable via yokes. Finally, a relationship between yokes and symplectic forms has been established in [[#References|[a4]]].
  
<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/y/y110/y110020/y11002018.png" /></td> </tr></table>
+
In order to define a yoke, let  $  M $
 +
be a smooth  $  d $-
 +
dimensional [[Manifold|manifold]] and let  $  \omega = ( \omega  ^ {1} \dots \omega  ^ {d} ) $
 +
and, correspondingly,  $  ( \omega, \omega  ^  \prime  ) = ( \omega  ^ {1} \dots \omega  ^ {d} , \omega ^ {\prime 1 } \dots \omega ^ {\prime d } ) $
 +
denote local coordinates on  $  M $
 +
and  $  M \times M $,
 +
respectively. Arbitrary components of  $  \omega $
 +
will be denoted by the letters  $  i,j,k,m, \dots $.
 +
For two sets of indices  $  K _ {t} = k _ {1} \dots k _ {t} $
 +
and  $  M _ {u} = m _ {1} \dots m _ {u} $
 +
and a smooth function  $  g : {M \times M } \rightarrow \mathbf R $,
 +
the symbol  $  /g _ {K _ {t}  ;M _ {u} } $
 +
is used for the values of the function
  
evaluated at the diagonal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002019.png" />, i.e.
+
$$
 +
g _ {K _ {t}  ;M _ {u} } ( \omega, \omega  ^  \prime  ) = {
 +
\frac{\partial  ^ {t + u } g ( \omega, \omega  ^  \prime  ) }{\partial  \omega ^ {k _ {1} } \dots \partial  \omega ^ {k _ {t} } \partial  \omega ^ {\prime m _ {1} } \dots \partial  \omega ^ {\prime m _ {u} } }
 +
}
 +
$$
  
<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/y/y110/y110020/y11002020.png" /></td> </tr></table>
+
evaluated at the diagonal of  $  M \times M $,
 +
i.e.
  
With this notation, a yoke is a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002021.png" />, such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002022.png" />:
+
$$
 +
/g _ {K _ {t}  ;M _ {u} } ( \omega ) = g _ {K _ {t}  ;M _ {u} } ( \omega, \omega ) .
 +
$$
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002023.png" />;
+
With this notation, a yoke is a smooth function  $  g : {M \times M } \rightarrow R $,
 +
such that for every  $  \omega \in M $:
  
ii) the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002024.png" /> is non-singular.
+
i)  $  /g _ {i; }  ( \omega ) = 0 $;
  
A normalized yoke is a yoke satisfying the additional condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002025.png" />. For any yoke <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002026.png" /> there exists a corresponding normalized yoke <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002027.png" />, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002028.png" />, and a dual yoke <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002029.png" />, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002030.png" />.
+
ii) the matrix  $  [ /g _ {i;j }  ( \omega ) ] $
 +
is non-singular.
  
In the statistical context the two most important examples of normalized yokes are the expected and the observed likelihood yoke. For a parametric statistical model with parameter space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002031.png" />, sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002032.png" /> and log-likelihood function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002033.png" />, the expected likelihood yoke is given by
+
A normalized yoke is a yoke satisfying the additional condition  $  g ( \omega, \omega ) = 0 $.  
 +
For any yoke  $  g $
 +
there exists a corresponding normalized yoke  $  {\overline{g}\; } $,
 +
given by  $  {\overline{g}\; } ( \omega, \omega  ^  \prime  ) = g ( \omega, \omega  ^  \prime  ) - g ( \omega  ^  \prime  , \omega  ^  \prime  ) $,  
 +
and a dual yoke $  g  ^ {*} $,
 +
given by $  g  ^ {*} ( \omega, \omega  ^  \prime  ) = {\overline{g}\; } ( \omega  ^  \prime  , \omega ) $.
  
<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/y/y110/y110020/y11002034.png" /></td> </tr></table>
+
In the statistical context the two most important examples of normalized yokes are the expected and the observed likelihood yoke. For a parametric statistical model with parameter space  $  M $,
 +
sample space  $  {\mathcal X} $
 +
and log-likelihood function  $  l : {M \times {\mathcal X} } \rightarrow \mathbf R $,
 +
the expected likelihood yoke is given by
 +
 
 +
$$
 +
g ( \omega, \omega  ^  \prime  ) = {\mathsf E} _ {\omega  ^  \prime  } \{ l ( \omega ;x ) - l ( \omega  ^  \prime  ;x ) \} .
 +
$$
  
 
The observed likelihood yoke is given by
 
The observed likelihood yoke is given by
  
<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/y/y110/y110020/y11002035.png" /></td> </tr></table>
+
$$
 +
g ( \omega, \omega  ^  \prime  ) = l ( \omega ; \omega  ^  \prime  ,a ) - l ( \omega  ^  \prime  ; \omega  ^  \prime  ,a ) .
 +
$$
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002036.png" /> is an auxiliary statistic such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002038.png" /> denotes the maximum-likelihood estimator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002039.png" /> (cf. also [[Maximum-likelihood method|Maximum-likelihood method]]), is bijective. Further examples of statistical yokes are related to contrast functions, see [[#References|[a5]]].
+
Here, $  a $
 +
is an auxiliary statistic such that the function $  x \rightarrow ( {\widehat \omega  } ,a ) $,  
 +
where $  {\widehat \omega  } $
 +
denotes the maximum-likelihood estimator of $  \omega $(
 +
cf. also [[Maximum-likelihood method|Maximum-likelihood method]]), is bijective. Further examples of statistical yokes are related to contrast functions, see [[#References|[a5]]].
  
Some further notation is needed for the discussion of properties of yokes. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002040.png" /> is a smooth function, one sets
+
Some further notation is needed for the discussion of properties of yokes. If $  f : M \rightarrow \mathbf R $
 +
is a smooth function, one sets
  
<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/y/y110/y110020/y11002041.png" /></td> </tr></table>
+
$$
 +
f _ {/K _ {t}  } = {
 +
\frac{\partial  ^ {t} f ( \omega ) }{\partial  \omega ^ {k _ {1} } \dots \partial  \omega ^ {k _ {t} } }
 +
} .
 +
$$
  
Furthermore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002042.png" /> is an alternative set of local coordinates for which arbitrary components are denoted by the letters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002043.png" /> and if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002044.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002046.png" /> are two sets of indices related to the local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002048.png" />, respectively, one sets
+
Furthermore, if $  \psi = ( \psi  ^ {1} \dots \psi  ^ {d} ) $
 +
is an alternative set of local coordinates for which arbitrary components are denoted by the letters $  a,b,c,d, \dots $
 +
and if for $  t, \tau = 1,2, \dots $
 +
$  C _ {t} $
 +
and $  K _  \tau  $
 +
are two sets of indices related to the local coordinates $  \psi $
 +
and $  \omega $,  
 +
respectively, one sets
  
<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/y/y110/y110020/y11002049.png" /></td> </tr></table>
+
$$
 +
\omega _ {C _ {t}  } ^ {K _  \tau  } = \sum _ {C _ {t} / \tau } \omega _ {/C _ {t1 }  } ^ {k _ {1} } \dots \omega _ {/C _ {t \tau }  } ^ {k _  \tau  } .
 +
$$
  
Here, the summation is over ordered partitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002050.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002051.png" /> (non-empty) subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002052.png" /> such that the order of the indices in each of the subsets is the same as the order within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002053.png" /> and such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002054.png" /> the first index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002055.png" /> comes before the first index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002056.png" /> as compared with the ordering within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002057.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002058.png" />, the sum is to be interpreted as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002059.png" />.
+
Here, the summation is over ordered partitions of $  C _ {t} = c _ {1} \dots c _ {t} $
 +
into $  \tau $(
 +
non-empty) subsets $  C _ {t1 }  \dots C _ {t \tau }  $
 +
such that the order of the indices in each of the subsets is the same as the order within $  C _ {t} $
 +
and such that for $  \mu = 1 \dots \tau - 1 $
 +
the first index of $  C _ {t \mu }  $
 +
comes before the first index of $  C _ {t, \mu + 1 }  $
 +
as compared with the ordering within $  C _ {t} $.  
 +
For $  \tau > t $,  
 +
the sum is to be interpreted as 0 $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002060.png" /> be an arbitrary yoke and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002061.png" />. Then the most important properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002062.png" /> are:
+
Let $  g $
 +
be an arbitrary yoke and let $  /g _ {;} = \{ {/g _ {K _ {t}  ;M _ {u} } } : {t,u = 1,2, \dots } \} $.  
 +
Then the most important properties of $  g $
 +
are:
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002063.png" /> satisfies the balance relation
+
a) /g _ {;} $
 +
satisfies the balance relation
  
<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/y/y110/y110020/y11002064.png" /></td> </tr></table>
+
$$
 +
/g _ {K _ {t}  ; } + \sum _ {K _ {t} /2 } /g _ {K _ {t1 }  ;K _ {t2 }  } = 0.
 +
$$
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002065.png" /> is a double derivative string, i.e. the transformation law is
+
b) /g _ {;} $
 +
is a double derivative string, i.e. the transformation law is
  
<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/y/y110/y110020/y11002066.png" /></td> </tr></table>
+
$$
 +
/g _ {C _ {t}  ;D _ {u} } = \sum _ {\tau = 1 } ^ { t }  \sum _ {\nu = 1 } ^ { u }  /g _ {K _  \tau  ;M _  \nu  } \omega _ {/C _ {t}  } ^ {K _  \tau  } \omega _ {/D _ {u}  } ^ {M _  \nu  } .
 +
$$
  
In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002067.png" /> is a symmetric non-singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002068.png" />-tensor, and consequently <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002069.png" /> equipped with this metric is a [[Riemannian manifold|Riemannian manifold]]. The inverse of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002070.png" /> will be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002071.png" />.
+
In particular, /g _ {i;j }  $
 +
is a symmetric non-singular $  ( 0,2 ) $-
 +
tensor, and consequently $  M $
 +
equipped with this metric is a [[Riemannian manifold|Riemannian manifold]]. The inverse of the matrix $  [ /g _ {i;j }  ] $
 +
will be denoted by $  [ /g ^ {i;j } ] $.
  
c) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002072.png" /> the collection of arrays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002073.png" />, where
+
c) For $  \alpha \in \mathbf R $
 +
the collection of arrays $  {\Gamma ^  \alpha  } = \{ { {\Gamma ^  \alpha  } {} _ {K _ {t}  }  ^ {i} } : {t = 1,2, \dots } \} $,  
 +
where
  
<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/y/y110/y110020/y11002074.png" /></td> </tr></table>
+
$$
 +
{\Gamma ^  \alpha  } {} _ {K _ {t}  }  ^ {i} = \left \{ {
 +
\frac{1 + \alpha }{2}
 +
} /g _ {K _ {t}  ;j } + {
 +
\frac{1 - \alpha }{2}
 +
} /g _ {j;K _ {t}  } \right \} /g ^ {i;j }
 +
$$
  
is a connection string, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002075.png" /> satisfies the transformation law
+
is a connection string, i.e. $  {\Gamma ^  \alpha  } $
 +
satisfies the transformation law
  
<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/y/y110/y110020/y11002076.png" /></td> </tr></table>
+
$$
 +
{\Gamma ^  \alpha  } {} _ {C _ {t}  }  ^ {a} = \left \{ \sum _ {\tau = 1 } ^ { t }  {\Gamma ^  \alpha  } {} _ {K _  \tau  }  ^ {i} \omega _ {/C _ {t}  } ^
 +
{K _  \tau  } \right \} \psi _ {/i }  ^ {a} .
 +
$$
  
In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002077.png" /> is the (upper) [[Christoffel symbol|Christoffel symbol]] of a torsion-free [[Affine connection|affine connection]], the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002079.png" />-connection, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002080.png" /> corresponding to the yoke <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002081.png" />.
+
In particular, $  {\Gamma ^  \alpha  } {} _ {k _ {1}  k _ {2} }  ^ {i} $
 +
is the (upper) [[Christoffel symbol|Christoffel symbol]] of a torsion-free [[Affine connection|affine connection]], the so-called $  \alpha $-
 +
connection, $  {\nabla ^  \alpha  } $
 +
corresponding to the yoke $  g $.
  
The expected and observed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002084.png" />-geometries, see [[#References|[a1]]] and [[#References|[a2]]], are those corresponding to the expected and observed likelihood yokes, respectively.
+
The expected and observed $  \alpha $-
 +
geometries, see [[#References|[a1]]] and [[#References|[a2]]], are those corresponding to the expected and observed likelihood yokes, respectively.
  
d) For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002085.png" /> there exists a sequence of tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002086.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002087.png" /> is a covariant tensor of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002088.png" />. The quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002089.png" /> are referred to as the tensorial components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002090.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002091.png" /> and are obtained by intertwining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002093.png" />, i.e. determined recursively by the equations
+
d) For $  \alpha \in \mathbf R $
 +
there exists a sequence of tensors $  {T ^  \alpha  } _ {;} = \{ { {T ^  \alpha  } _ {I _  \tau  ;J _  \upsilon  } } : {\tau, \upsilon = 1,2, \dots } \} , $
 +
such that $  {T ^  \alpha  } _ {I _  \tau  ;J _  \upsilon  } $
 +
is a covariant tensor of degree $  \tau + \upsilon $.  
 +
The quantities $  {T ^  \alpha  } _ {;} $
 +
are referred to as the tensorial components of /g _ {;} $
 +
with respect to $  {\Gamma ^  \alpha  } $
 +
and are obtained by intertwining /g _ {;} $
 +
and $  {\Gamma ^  \alpha  } $,  
 +
i.e. determined recursively by the equations
  
<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/y/y110/y110020/y11002094.png" /></td> </tr></table>
+
$$
 +
/g _ {K _ {t}  ;M _ {u} } = \sum _ {\tau = 1 } ^ { t }  \sum _ {\nu = 1 } ^ { u }  {T ^  \alpha  } _ {I _  \tau  ;J _  \upsilon  } {\Gamma ^  \alpha  } {} _ {K _ {t}  } ^ {I _  \tau  } {\Gamma ^  \alpha  } {} _ {M _ {u}  } ^ {J _  \upsilon  } ,
 +
$$
  
 
where
 
where
  
<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/y/y110/y110020/y11002095.png" /></td> </tr></table>
+
$$
 +
{\Gamma ^  \alpha  } {} _ {K _ {t}  } ^ {I _  \tau  } = \sum _ {K _ {t} / \tau } {\Gamma ^  \alpha  } {} _ {K _ {t1 }  } ^ {i _ {1} } \dots {\Gamma ^  \alpha  } {} _ {K _ {t \tau }  } ^ {i _  \tau  } .
 +
$$
  
In terms of the local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002096.png" />, an invariant Taylor expansion, around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002097.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002098.png" />, of a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y11002099.png" /> is of the form
+
In terms of the local coordinates $  \omega $,  
 +
an invariant Taylor expansion, around $  m \in M $
 +
or $  \omega  ^  \prime  = \omega  ^  \prime  ( m ) $,  
 +
of a smooth function $  f $
 +
is of the form
  
<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/y/y110/y110020/y110020100.png" /></td> </tr></table>
+
$$
 +
f ( \omega ) = f ( \omega  ^  \prime  ) + \sum _ {\tau = 1 } ^  \infty  {
 +
\frac{1}{\tau ! }
 +
} {f ^ { 1 }  } _ {//I _  \tau  } ( \omega  ^  \prime  ) \gamma ^ {I _  \tau  } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020101.png" /> are the tensorial components of the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020102.png" /> with respect to the connection string <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020103.png" /> given recursively by
+
where $  \{ { {f ^ { 1 }  } _ {//I _  \tau  } } : {\tau = 1,2, \dots } \} $
 +
are the tensorial components of the derivatives $  \{ {f _ {/K _ {t}  } } : {\tau = 1,2, \dots } \} $
 +
with respect to the connection string $  {\Gamma ^ { 1 }  } $
 +
given recursively by
  
<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/y/y110/y110020/y110020104.png" /></td> </tr></table>
+
$$
 +
f _ {/K _ {t}  } = \sum _ {\tau = 1 } ^ { t }  {f ^ { 1 }  } _ {//I _  \tau  } {\Gamma ^ { 1 }  } {} _ {K _ {t}  } ^ {I _  \tau  } .
 +
$$
  
Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020105.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020106.png" /> indicates the extended normal coordinates around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020107.png" /> whose components are given by
+
Furthermore, $  \gamma ^ {I _  \tau  } = \gamma ^ {i _ {1} } \dots \gamma ^ {i _  \tau  } $,  
 +
where $  \gamma $
 +
indicates the extended normal coordinates around $  m $
 +
whose components are given by
  
<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/y/y110/y110020/y110020108.png" /></td> </tr></table>
+
$$
 +
\gamma  ^ {i} ( \omega ) = {\overline{g}\; } _ {;j }  ( \omega, \omega  ^  \prime  ) /g ^ {i;j } ,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020109.png" /> being the normalized yoke corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020111.png" />.
+
$  {\overline{g}\; } $
 +
being the normalized yoke corresponding to $  g $
 +
and $  \omega  ^  \prime  = \omega  ^  \prime  ( m ) $.
  
The Taylor expansion is invariant in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y110/y110020/y110020113.png" /> are tensors.
+
The Taylor expansion is invariant in the sense that $  {f ^ { 1 }  } _ {//I _  \tau  } $
 +
and $  \gamma ^ {I _  \tau  } $
 +
are tensors.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S-I. Amari,  "Differential-geometrical methods in statistics" , ''Lecture Notes in Statistics'' , '''28''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O.E. Barndorff-Nielsen,  "Likelihood and observed geometries"  ''Ann. Stat.'' , '''14'''  (1986)  pp. 856–873</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O.E. Barndorff-Nielsen,  "Differential geometry and statistics. Some mathematical aspects"  ''Indian J. Math. (Ramanujan Centenary Volume)'' , '''29'''  (1987)  pp. 335–350</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O.E. Barndorff-Nielsen,  P.E Jupp,  "Statistics, yokes and symplectic geometry"  ''Ann. Toulouse'' , '''to appear'''  (1997)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Blæsild,  "Yokes and tensors derived from yokes"  ''Ann. Inst. Statist. Math.'' , '''43'''  (1991)  pp. 95–113</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S-I. Amari,  "Differential-geometrical methods in statistics" , ''Lecture Notes in Statistics'' , '''28''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O.E. Barndorff-Nielsen,  "Likelihood and observed geometries"  ''Ann. Stat.'' , '''14'''  (1986)  pp. 856–873</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O.E. Barndorff-Nielsen,  "Differential geometry and statistics. Some mathematical aspects"  ''Indian J. Math. (Ramanujan Centenary Volume)'' , '''29'''  (1987)  pp. 335–350</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O.E. Barndorff-Nielsen,  P.E Jupp,  "Statistics, yokes and symplectic geometry"  ''Ann. Toulouse'' , '''to appear'''  (1997)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Blæsild,  "Yokes and tensors derived from yokes"  ''Ann. Inst. Statist. Math.'' , '''43'''  (1991)  pp. 95–113</TD></TR></table>

Latest revision as of 12:10, 11 February 2020


The concept of a yoke, introduced in [a3], is of great importance in relation to geometric, i.e. parametrization invariant, calculations on statistical models (cf. also Differential geometry in statistical inference; Statistical manifold). A yoke on a model $ M $ induces a metric and families of connections, derivative strings and tensors on $ M $ in terms of which geometric properties of $ M $ may be formulated, see [a5]. Differences and similarities between the expected and observed geometry of $ M $ may be discussed using yokes, see [a5]. Furthermore, invariant Taylor expansions of functions defined on $ M $ are obtainable via yokes. Finally, a relationship between yokes and symplectic forms has been established in [a4].

In order to define a yoke, let $ M $ be a smooth $ d $- dimensional manifold and let $ \omega = ( \omega ^ {1} \dots \omega ^ {d} ) $ and, correspondingly, $ ( \omega, \omega ^ \prime ) = ( \omega ^ {1} \dots \omega ^ {d} , \omega ^ {\prime 1 } \dots \omega ^ {\prime d } ) $ denote local coordinates on $ M $ and $ M \times M $, respectively. Arbitrary components of $ \omega $ will be denoted by the letters $ i,j,k,m, \dots $. For two sets of indices $ K _ {t} = k _ {1} \dots k _ {t} $ and $ M _ {u} = m _ {1} \dots m _ {u} $ and a smooth function $ g : {M \times M } \rightarrow \mathbf R $, the symbol $ /g _ {K _ {t} ;M _ {u} } $ is used for the values of the function

$$ g _ {K _ {t} ;M _ {u} } ( \omega, \omega ^ \prime ) = { \frac{\partial ^ {t + u } g ( \omega, \omega ^ \prime ) }{\partial \omega ^ {k _ {1} } \dots \partial \omega ^ {k _ {t} } \partial \omega ^ {\prime m _ {1} } \dots \partial \omega ^ {\prime m _ {u} } } } $$

evaluated at the diagonal of $ M \times M $, i.e.

$$ /g _ {K _ {t} ;M _ {u} } ( \omega ) = g _ {K _ {t} ;M _ {u} } ( \omega, \omega ) . $$

With this notation, a yoke is a smooth function $ g : {M \times M } \rightarrow R $, such that for every $ \omega \in M $:

i) $ /g _ {i; } ( \omega ) = 0 $;

ii) the matrix $ [ /g _ {i;j } ( \omega ) ] $ is non-singular.

A normalized yoke is a yoke satisfying the additional condition $ g ( \omega, \omega ) = 0 $. For any yoke $ g $ there exists a corresponding normalized yoke $ {\overline{g}\; } $, given by $ {\overline{g}\; } ( \omega, \omega ^ \prime ) = g ( \omega, \omega ^ \prime ) - g ( \omega ^ \prime , \omega ^ \prime ) $, and a dual yoke $ g ^ {*} $, given by $ g ^ {*} ( \omega, \omega ^ \prime ) = {\overline{g}\; } ( \omega ^ \prime , \omega ) $.

In the statistical context the two most important examples of normalized yokes are the expected and the observed likelihood yoke. For a parametric statistical model with parameter space $ M $, sample space $ {\mathcal X} $ and log-likelihood function $ l : {M \times {\mathcal X} } \rightarrow \mathbf R $, the expected likelihood yoke is given by

$$ g ( \omega, \omega ^ \prime ) = {\mathsf E} _ {\omega ^ \prime } \{ l ( \omega ;x ) - l ( \omega ^ \prime ;x ) \} . $$

The observed likelihood yoke is given by

$$ g ( \omega, \omega ^ \prime ) = l ( \omega ; \omega ^ \prime ,a ) - l ( \omega ^ \prime ; \omega ^ \prime ,a ) . $$

Here, $ a $ is an auxiliary statistic such that the function $ x \rightarrow ( {\widehat \omega } ,a ) $, where $ {\widehat \omega } $ denotes the maximum-likelihood estimator of $ \omega $( cf. also Maximum-likelihood method), is bijective. Further examples of statistical yokes are related to contrast functions, see [a5].

Some further notation is needed for the discussion of properties of yokes. If $ f : M \rightarrow \mathbf R $ is a smooth function, one sets

$$ f _ {/K _ {t} } = { \frac{\partial ^ {t} f ( \omega ) }{\partial \omega ^ {k _ {1} } \dots \partial \omega ^ {k _ {t} } } } . $$

Furthermore, if $ \psi = ( \psi ^ {1} \dots \psi ^ {d} ) $ is an alternative set of local coordinates for which arbitrary components are denoted by the letters $ a,b,c,d, \dots $ and if for $ t, \tau = 1,2, \dots $ $ C _ {t} $ and $ K _ \tau $ are two sets of indices related to the local coordinates $ \psi $ and $ \omega $, respectively, one sets

$$ \omega _ {C _ {t} } ^ {K _ \tau } = \sum _ {C _ {t} / \tau } \omega _ {/C _ {t1 } } ^ {k _ {1} } \dots \omega _ {/C _ {t \tau } } ^ {k _ \tau } . $$

Here, the summation is over ordered partitions of $ C _ {t} = c _ {1} \dots c _ {t} $ into $ \tau $( non-empty) subsets $ C _ {t1 } \dots C _ {t \tau } $ such that the order of the indices in each of the subsets is the same as the order within $ C _ {t} $ and such that for $ \mu = 1 \dots \tau - 1 $ the first index of $ C _ {t \mu } $ comes before the first index of $ C _ {t, \mu + 1 } $ as compared with the ordering within $ C _ {t} $. For $ \tau > t $, the sum is to be interpreted as $ 0 $.

Let $ g $ be an arbitrary yoke and let $ /g _ {;} = \{ {/g _ {K _ {t} ;M _ {u} } } : {t,u = 1,2, \dots } \} $. Then the most important properties of $ g $ are:

a) $ /g _ {;} $ satisfies the balance relation

$$ /g _ {K _ {t} ; } + \sum _ {K _ {t} /2 } /g _ {K _ {t1 } ;K _ {t2 } } = 0. $$

b) $ /g _ {;} $ is a double derivative string, i.e. the transformation law is

$$ /g _ {C _ {t} ;D _ {u} } = \sum _ {\tau = 1 } ^ { t } \sum _ {\nu = 1 } ^ { u } /g _ {K _ \tau ;M _ \nu } \omega _ {/C _ {t} } ^ {K _ \tau } \omega _ {/D _ {u} } ^ {M _ \nu } . $$

In particular, $ /g _ {i;j } $ is a symmetric non-singular $ ( 0,2 ) $- tensor, and consequently $ M $ equipped with this metric is a Riemannian manifold. The inverse of the matrix $ [ /g _ {i;j } ] $ will be denoted by $ [ /g ^ {i;j } ] $.

c) For $ \alpha \in \mathbf R $ the collection of arrays $ {\Gamma ^ \alpha } = \{ { {\Gamma ^ \alpha } {} _ {K _ {t} } ^ {i} } : {t = 1,2, \dots } \} $, where

$$ {\Gamma ^ \alpha } {} _ {K _ {t} } ^ {i} = \left \{ { \frac{1 + \alpha }{2} } /g _ {K _ {t} ;j } + { \frac{1 - \alpha }{2} } /g _ {j;K _ {t} } \right \} /g ^ {i;j } $$

is a connection string, i.e. $ {\Gamma ^ \alpha } $ satisfies the transformation law

$$ {\Gamma ^ \alpha } {} _ {C _ {t} } ^ {a} = \left \{ \sum _ {\tau = 1 } ^ { t } {\Gamma ^ \alpha } {} _ {K _ \tau } ^ {i} \omega _ {/C _ {t} } ^ {K _ \tau } \right \} \psi _ {/i } ^ {a} . $$

In particular, $ {\Gamma ^ \alpha } {} _ {k _ {1} k _ {2} } ^ {i} $ is the (upper) Christoffel symbol of a torsion-free affine connection, the so-called $ \alpha $- connection, $ {\nabla ^ \alpha } $ corresponding to the yoke $ g $.

The expected and observed $ \alpha $- geometries, see [a1] and [a2], are those corresponding to the expected and observed likelihood yokes, respectively.

d) For $ \alpha \in \mathbf R $ there exists a sequence of tensors $ {T ^ \alpha } _ {;} = \{ { {T ^ \alpha } _ {I _ \tau ;J _ \upsilon } } : {\tau, \upsilon = 1,2, \dots } \} , $ such that $ {T ^ \alpha } _ {I _ \tau ;J _ \upsilon } $ is a covariant tensor of degree $ \tau + \upsilon $. The quantities $ {T ^ \alpha } _ {;} $ are referred to as the tensorial components of $ /g _ {;} $ with respect to $ {\Gamma ^ \alpha } $ and are obtained by intertwining $ /g _ {;} $ and $ {\Gamma ^ \alpha } $, i.e. determined recursively by the equations

$$ /g _ {K _ {t} ;M _ {u} } = \sum _ {\tau = 1 } ^ { t } \sum _ {\nu = 1 } ^ { u } {T ^ \alpha } _ {I _ \tau ;J _ \upsilon } {\Gamma ^ \alpha } {} _ {K _ {t} } ^ {I _ \tau } {\Gamma ^ \alpha } {} _ {M _ {u} } ^ {J _ \upsilon } , $$

where

$$ {\Gamma ^ \alpha } {} _ {K _ {t} } ^ {I _ \tau } = \sum _ {K _ {t} / \tau } {\Gamma ^ \alpha } {} _ {K _ {t1 } } ^ {i _ {1} } \dots {\Gamma ^ \alpha } {} _ {K _ {t \tau } } ^ {i _ \tau } . $$

In terms of the local coordinates $ \omega $, an invariant Taylor expansion, around $ m \in M $ or $ \omega ^ \prime = \omega ^ \prime ( m ) $, of a smooth function $ f $ is of the form

$$ f ( \omega ) = f ( \omega ^ \prime ) + \sum _ {\tau = 1 } ^ \infty { \frac{1}{\tau ! } } {f ^ { 1 } } _ {//I _ \tau } ( \omega ^ \prime ) \gamma ^ {I _ \tau } , $$

where $ \{ { {f ^ { 1 } } _ {//I _ \tau } } : {\tau = 1,2, \dots } \} $ are the tensorial components of the derivatives $ \{ {f _ {/K _ {t} } } : {\tau = 1,2, \dots } \} $ with respect to the connection string $ {\Gamma ^ { 1 } } $ given recursively by

$$ f _ {/K _ {t} } = \sum _ {\tau = 1 } ^ { t } {f ^ { 1 } } _ {//I _ \tau } {\Gamma ^ { 1 } } {} _ {K _ {t} } ^ {I _ \tau } . $$

Furthermore, $ \gamma ^ {I _ \tau } = \gamma ^ {i _ {1} } \dots \gamma ^ {i _ \tau } $, where $ \gamma $ indicates the extended normal coordinates around $ m $ whose components are given by

$$ \gamma ^ {i} ( \omega ) = {\overline{g}\; } _ {;j } ( \omega, \omega ^ \prime ) /g ^ {i;j } , $$

$ {\overline{g}\; } $ being the normalized yoke corresponding to $ g $ and $ \omega ^ \prime = \omega ^ \prime ( m ) $.

The Taylor expansion is invariant in the sense that $ {f ^ { 1 } } _ {//I _ \tau } $ and $ \gamma ^ {I _ \tau } $ are tensors.

References

[a1] S-I. Amari, "Differential-geometrical methods in statistics" , Lecture Notes in Statistics , 28 , Springer (1985)
[a2] O.E. Barndorff-Nielsen, "Likelihood and observed geometries" Ann. Stat. , 14 (1986) pp. 856–873
[a3] O.E. Barndorff-Nielsen, "Differential geometry and statistics. Some mathematical aspects" Indian J. Math. (Ramanujan Centenary Volume) , 29 (1987) pp. 335–350
[a4] O.E. Barndorff-Nielsen, P.E Jupp, "Statistics, yokes and symplectic geometry" Ann. Toulouse , to appear (1997)
[a5] P. Blæsild, "Yokes and tensors derived from yokes" Ann. Inst. Statist. Math. , 43 (1991) pp. 95–113
How to Cite This Entry:
Yoke. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Yoke&oldid=17982
This article was adapted from an original article by P. Blæsild (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article