View source for Yoke

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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  $  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]]].

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

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

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|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  $  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|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|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  $  \alpha $-
geometries, see [[#References|[a1]]] and [[#References|[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====
<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>