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Information geometry reached maturity through the work of Shun'ichi Amari and other Japanese mathematicians in the 1980s. Amari and Nagaoka's book, Methods of Information Geometry, is cited by most works of the relatively young field due to its broad coverage of significant developments attained using the methods of information geometry up to the year 2000. Many of these developments were previously only available in Japanese-language publications.
Define an n-set to be a set V with cardinality . To choose an element v (value, state, point, outcome) from an n-set V, one needs to specify b-sets (default b=2), if one disregards all but the cardinality. That is, nats of information are required to specify v; equivalently, bits are needed.
By considering the occurrences of values from , one has an alternate way to refer to , through . First, one chooses an occurrence , which requires information of bits. To specify v, one subtracts the excess information used to choose one from all those linked to , this is . Then, is the number of portions fitting into . Thus, one needs bits to choose one of them. So the information (variable size, code length, number of bits) needed to refer to , considering its occurrences in a message is
Finally, is the normalized portion of information needed to code all occurrences of one . The averaged code length over all values is . is called the entropy of a random variable.
With a probability distribution one looks at a variable through an observation context like a message or an experimental setup.
The context can often be identified by a set of parameters through combinatorial reasoning. The parameters can have an arbitrary number of dimensions and can be very local or less so, as long as the context given by a certain produces every value of , i.e. the support does not change as function of . Every determines one probability distribution for . Basically all distributions for which there exists an explicit analytical formula fall into this category (Binomial, Normal, Poisson, ...). The parameters in these cases have a concrete meaning in the underlying setup, which is a statistical model for the context of .
The parameters are quite different in nature from itself, because they do not describe , but the observation context for .
A parameterization of the form
that mixes different distributions , is called a mixture distribution, mixture or -parameterization or mixture for short. All such parameterizations are related through an affine transformation. A parameterization with such a transformation rule is called flat.
A flat parameterization for is an exponential or parameterization, because the parameters are in the exponent of . There are several important distributions, like Normal and Poisson, that fall into this category. These distributions are collectively referred to as exponential family or -family. The -manifold for such distributions is not affine, but the manifold is. This is called -affine. The parameterization for the exponential family can be mapped to the one above by making another parameter and extend .
Differential geometry applied to probability
In information geometry, the methods of differential geometry are applied to describe the space of probability distributions for one variable . This is done by using a coordinate or atlas. Furthermore, the probability must be a differentiable and invertible function of . In this case, the are coordinates of the -space, and the latter is a differential manifold.
Derivatives are defined as is usual for a differentiable manifold:
with , for a real-valued function on .
Given a function on , one may "geometrize" it by taking it to define a new manifold. This is done by defining coordinate functions on this new manifold as
In this way one "geometricizes" a function , by encoding it into the coordinates used to describe the system.
For the inverse is and the resulting manifold of points is called the -representation. The manifold itself is called the -representation. The - or -representations, in the sense used here, does not refer to the parameterization families of the distribution.
In ordinary differential geometry, there is no canonical coordinate system on the manifold; thus, typically, all discussion must be with regard to an atlas, that is, with regard to functions on the manifold. As a result, tangent spaces and vectors are defined as operators acting on this space of functions. So, for example, in ordinary differential geometry, the basis vectors of the tangent space are the operators .
However, with probability distributions , one can calculate value-wise. So it is possible to express a tangent space vector directly as ( -representation ) or ( -representation ), and not as operators.
Here are equivalent formulas of the Fisher information metric.
, the base vector in the -representation, is also called the score.
. This is the same for and families.
with mimimum for entails and is applied only to the first parameter, and only to the second. is the Kullback-Leibler divergence or relative entropy applicable to the -families.
For one has . is the Hellinger distance applicable to the -family. also evaluates to the Fisher metric.
This relation with a divergence will be revisited further down.
The Fisher metric is motivated by
it satisfying the requirements for an inner product
its invariance for a sufficient statistic deterministic mapping from one variable to another and more general for , i.e. a broadened distribution has smaller .
it being the Cramér–Rao bound. , therefore any satisfying belongs to .
For any one has , therefore . .
So and therefore . and with inefficient estimator one gets the Cramér–Rao bound .
Given vector fields and lying in the tangent bundle, the affine connection describes how to differentiate the vector field along the direction. It is itself a vector field; it is the sum of the infinitesimal change in the vector field , as one moves along the direction , plus the infinitesimal change of the vector due to its parallel transport along the direction . That is, it takes into account the changing nature of what it means to move a coordinate system in a "parallel" fashion, as one moves about in the manifold. In terms of the basis vectors , one has the components:
A non-metric connection is not determined by a metric tensor; instead, it is and restricted by the requirement that the parallel transport between points and must be a linear combination of the base vectors in . Here,
expresses the parallel transport of as linear combination of the base vectors in , i.e. the new minus the change. Note that it is not a tensor (does not transform as a tensor).
For such a metric, one can construct a dual connection to make
for parallel transport using and .
For the mentioned -families the affine connection is called the -connection and can also be expressed in more ways.
is a metric connection and with .
i.e. is dual to with respect to the Fisher metric.
If this is called -affine. Its dual is then -affine.
A function of two distributions (points) with minimum for entails and . is applied only to the first parameter, and only to the second. is the direction, which brought the two points to be equal, when applied to the first parameter, and to diverge again, when applied to the second parameter, i.e. . The sign cancels in , which we can define to be a metric , if always positive.
The absolute derivative of along yields candidates for dual connections . This metric and the connections relate to the Taylor series expansion for the first parameter or second parameter. Here for the first parameter:
The term is called the divergence or contrast function. A good choice is with convex for . From Jensen's inequality it follows that and, for , we have
which is the Kullback-Leibler divergence or relative entropy applicable to the -families. In the above,
is the Fisher metric. For a different yields
The Hellinger distance applicable to the -family is
In this case, also evaluates to the Fisher metric.
We now consider two manifolds and , represented by two sets of coordinate functions and . The corresponding tangent space basis vectors will be denoted by and . The bilinear map associates a quantity to the dual base vectors. This defines an affine connection for and affine connection for that keep constant for parallel transport of and , defined through and .
If is flat, then there exists a coordinate system , that does not change over . In order to keep constant, must not change either, i.e. is also flat. Furthermore, in this case, we can choose coordinate systems such that
If results as a function on , then making , both coordinate system function sets describe . The connections are such, though, that makes flat and makes flat. This dual space is denoted as .
Because of the linear transform between the flat coordinate systems, we have and .
Because and so for it is possible to define two potentials and through and ( Legendre transform ).These are and .
This naturally leads to the following definition of a canonical divergence:
Note the summation that is a representation of the metric due to .
The meaning of the canonical divergence depends on the meaning of the metric and vice versa ( ). For the metric (Fisher metric) with the dual connections this is the relative entropy. For the self-dual Euclidean space leads to
Similar to the Euclidean space the following holds:
Triangular relation: (just substitute )
If is not dually flat then this generalizes to:
The last part drops in case of dual flatness. is the exponential map.
Pythagorean Theorem: For and meeting on orthogonal lines at ( )
For and with a -autoparallel sub-manifold implies that the -geodesic connecting and is orthogonal to .
By projecting onto of a curve one can calculate
the divergence of the curve where
and with .
With this becomes .
For an autoparallel sub-manifold parallel transport in it can be expressed with the sub-manifold's base vectors, i.e. . A one-dimensional autoparallel sub-manifold is a geodesic.
Canonical divergence for the exponential family
For the exponential family one has . Applying on both sides yields . The other potential ( is entropy, and was used). is the covariance of , the Cramér–Rao bound, i.e. an efficient estimator must be exponential.
The canonical divergence is given by the Kullback-Leibler divergence and the triangulation is .
The minimal divergence to a sub-manifold given by a restriction like some constant means maximizing . With this corresponds to the maximum entropy principle.
Canonical divergence for general alpha families
For general -affine manifolds with one has:
The connection induced by the divergence is not flat unless . Then the Pythagorean theorem for two curves intersecting orthogonally at is:
Shun'ichi Amari, Hiroshi Nagaoka - Methods of information geometry, Translations of mathematical monographs; v. 191, American Mathematical Society, 2000 (ISBN 978-0821805312)
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N. N. Cencov - Statistical Decision Rules and Optimal Inference, Translations of Mathematical Monographs; v. 53, American Mathematical Society, 1982
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