In mathematics and especially in statistical inference, information geometry is the study of probability and information by way of differential geometry. It reached maturity through the work of Shun'ichi Amari in the 1980s, with what is currently the canonical reference book: Methods of information geometry.
Contents 
The main tenet of information geometry is that many important structures in probability theory, information theory and statistics can be treated as structures in differential geometry by regarding a space of probabilities as a differentiable manifold endowed with a Riemannian metric and a family of affine connections distinct from the canonical affine connection. The eaffine connection and maffine connection geometrize expectation and maximization, as in the expectationmaximization algorithm.
For example,
The importance of studying statistical structures as geometrical structures lies in the fact that geometric structures are invariant under coordinate transforms. For example, a family of probability distributions, such as Gaussian distributions, may be transformed into another family of distributions, such as lognormal distributions, by a change of variables. However, the fact of it being an exponential family is not changed, since the latter is a geometric property. The distance between two distributions in this family defined through Fisher metric will also be preserved.
The statistician Fisher recognized in the 1920s that there is an intrinsic measure of amount of information for statistical estimators. The Fisher information matrix was shown by Cramer and Rao to be a Riemannian metric on the space of probabilities, and became known as Fisher information metric.
The mathematician Cencov (Chentsov) proved in the 1960s and 1970s that on the space of probability distributions on a sample space containing at least three points,
Both of these uniqueness are, of course, up to the multiplication by a constant.
Amari and Nagaoka's study in the 1980s brought all these results together, with the introduction of the concept of dualaffine connections, and the interplay among metric, affine connection and divergence. In particular,
Also, Amari and Kumon showed that asymptotic efficiency of estimates and tests can be represented by geometrical quantities.
Information geometry is based primarily on the Fisher information metric:
Substituting i = −log(p) from information theory, the formula becomes:
The history of information geometry is associated with the discoveries of at least the following people, and many others
An important concept in information geometry is the natural gradient. The concept and theory of the natural gradient suggests an adjustment to the energy function of a learning rule. This adjustment takes into account the curvature of the (prior) statistical differential manifold, by way of the Fisher information metric.
This concept has many important applications in blind signal separation, neural networks, artificial intelligence, and other engineering problems that deal with information. Experimental results have shown that application of the concept leads to substantial performance gains.
Other applications concern statistics of stochastic processes and approximate finite dimensional solutions of the filtering problem (stochastic processes). As the nonlinear filtering problem admits an infinite dimensional solution in general, one can use a geometric structure in the space of probability distributions to project the infinite dimensional filter into an approximate finite dimensional one, leading to the projection filters introduced in 1987 by Bernard Hanzon.
