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