Yoke
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 induces a metric and families of connections, derivative strings and tensors on
in terms of which geometric properties of
may be formulated, see [a5]. Differences and similarities between the expected and observed geometry of
may be discussed using yokes, see [a5]. Furthermore, invariant Taylor expansions of functions defined on
are obtainable via yokes. Finally, a relationship between yokes and symplectic forms has been established in [a4].
In order to define a yoke, let be a smooth
-dimensional manifold and let
and, correspondingly,
denote local coordinates on
and
, respectively. Arbitrary components of
will be denoted by the letters
. For two sets of indices
and
and a smooth function
, the symbol
is used for the values of the function
![]() |
evaluated at the diagonal of , i.e.
![]() |
With this notation, a yoke is a smooth function , such that for every
:
i) ;
ii) the matrix is non-singular.
A normalized yoke is a yoke satisfying the additional condition . For any yoke
there exists a corresponding normalized yoke
, given by
, and a dual yoke
, given by
.
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 , sample space
and log-likelihood function
, the expected likelihood yoke is given by
![]() |
The observed likelihood yoke is given by
![]() |
Here, is an auxiliary statistic such that the function
, where
denotes the maximum-likelihood estimator of
(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 is a smooth function, one sets
![]() |
Furthermore, if is an alternative set of local coordinates for which arbitrary components are denoted by the letters
and if for
and
are two sets of indices related to the local coordinates
and
, respectively, one sets
![]() |
Here, the summation is over ordered partitions of into
(non-empty) subsets
such that the order of the indices in each of the subsets is the same as the order within
and such that for
the first index of
comes before the first index of
as compared with the ordering within
. For
, the sum is to be interpreted as
.
Let be an arbitrary yoke and let
. Then the most important properties of
are:
a) satisfies the balance relation
![]() |
b) is a double derivative string, i.e. the transformation law is
![]() |
In particular, is a symmetric non-singular
-tensor, and consequently
equipped with this metric is a Riemannian manifold. The inverse of the matrix
will be denoted by
.
c) For the collection of arrays
, where
![]() |
is a connection string, i.e. satisfies the transformation law
![]() |
In particular, is the (upper) Christoffel symbol of a torsion-free affine connection, the so-called
-connection,
corresponding to the yoke
.
The expected and observed -geometries, see [a1] and [a2], are those corresponding to the expected and observed likelihood yokes, respectively.
d) For there exists a sequence of tensors
such that
is a covariant tensor of degree
. The quantities
are referred to as the tensorial components of
with respect to
and are obtained by intertwining
and
, i.e. determined recursively by the equations
![]() |
where
![]() |
In terms of the local coordinates , an invariant Taylor expansion, around
or
, of a smooth function
is of the form
![]() |
where are the tensorial components of the derivatives
with respect to the connection string
given recursively by
![]() |
Furthermore, , where
indicates the extended normal coordinates around
whose components are given by
![]() |
being the normalized yoke corresponding to
and
.
The Taylor expansion is invariant in the sense that and
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 |
Yoke. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Yoke&oldid=17982