| Information Geometry |
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| CATEGORIES ABOUT INFORMATION GEOMETRY | |
| differential geometry | |
| information theory | |
| probability theory | |
| statistics | |
| category theory | |
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INTRODUCTION 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 Differential Manifold endowed with a Riemannian Metric and a family of Affine Connection s. 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 log-normal 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 great 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's study in the 1980s brought all these results together, with the introduction of the concept of Dual-affine Connection s, and the interplay among Metric , Affine Connection and Divergence(information Geometry) . In particular,
BASIC CONCEPTS
FISHER INFORMATION METRIC AS A RIEMANNIAN METRIC Information geometry is based primarily on the Fisher Information Metric : : Substituting ''i'' = −log(''p'') from Information Theory , the formula becomes: : Which can be thought of intuitively as: "The distance between two points on a statistical differential manifold is the amount of information between them, i.e. the informational difference between them." Thus, if a point in information space represents the state of a system, then the trajectory of that point will, on average, be a Random Walk through information space, i.e. will Diffuse according to Brownian Motion . With this in mind, the information space can be thought of as a Fitness Landscape , a trajectory through this space being an "evolution". The Brownian motion of evolution trajectories thus represents the No Free Lunch phenomena discussed by Stuart Kauffman . HISTORY The history of information is associated with the discoveries of at least the following people, and many others
SOME APPLICATIONS Natural gradient 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 Network s, Artificial Intelligence , and other engineering problems that deal with information. Experimental results have shown that application of the concept leads to substantial performance gains. REFERENCES
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