nLab
information geometry

Idea

Information geometry aims to apply the techniques of differential geometry to statistics. Often it is useful to think of a family of probability distributions as a statistical manifold. For example, normal Gaussian distributions form a 2-dimensional manifold, parameterised by $(\mu, \sigma)$, mean and standard deviation. On such manifolds there are notions of Riemannian metric, connection, curvature, and so on, of statistical relevance.

More precisely,

Kullback-Leibler information, or relative entropy, features as a measure of divergence (not quite a metric, because it’s asymmetric), and Fisher information takes the role of curvature. One useful aspect of information geometry is that it gives a means to prove results about statistical models, simply by considering them as well-behaved geometrical objects. For instance, it’s basically a tautology to say that a manifold is not changing much in the vicinity of points of low curvature, and changing greatly near points of high curvature. Stated more precisely, and then translated back into probabilistic language, this becomes the Cramer-Rao inequality, that the variance of a parameter estimator is at least the reciprocal of the Fisher information. (Shalizi)

Founders of the systematical theory are N. N. Chentsov and Shun-ichi Amari.

Definitions

For $X$ a measurable space let $S$ be (a subspace of) the space of probability measures on $X$, equipped with the structure of a smooth manifold.

The Fisher metric on $S$ is the Riemannian metric given on two vector fields $v,w \in T S$ by

$$g(v,w)_s := E_s( v(log s) w(log s)) \,,$$

where $E_s(\cdots)$ denotes the expectation value under the measure $s \in S$ of the function $x \mapsto v(log s)_x w(log s)_x$ on $X$.

For instance (Amari, Section 2.1).

Related concepts

• entropy
• Fisher metric
• quantum information
• relative entropy, also known as the Kullback-Leibler divergence

References

See also Fisher metric, where Fisher metric in other contexts and quantum generalizations are treated. See also quantum information.

Textbooks providing the big picture

• Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachhöfer, Information geometry, Ergeb. der Mathematik and ihrer Grenzgebiete 3. Folge, 64, Springer 2017
• Shun-ichi Amari, O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen, and C. R. Rao, Differential geometry in statistical inference (project Euclid)

• Shun-ichi Amari, Information geometry and its applications, Applied Mathematical Sciences 194, Springer 2016
• Shun-ichi Amari, Hiroshi Nagaoka, Methods of information geometry, Transactions of mathematical monographs; v. 191, American Mathematical Society, 2000.

For a series of articles, see

• John Baez, Information Geometry

Lecture notes include

• Shun-ichi Amari, MaxEnt 2007 lecture
• Shun-ichi Amari, ETVC08 lecture.
• Sanjoy Dasgupta, lecture.

See also

• Hông Vân Lê, Statistical manifolds are statistical models, Journal of Geometry 84(1-2), March 2006, pp. 83-93.

• Blog post.

A brief introduction with more references is

• Cosma Shalizi, Information geometry (webpage)

Several people have noted an equivalence between statistic inference as parametric model selection and statistical mechanics on a statistical manifold, e.g.,

• Vijay Balasubramanian, Statistical Inference, Occam’s Razor and Statistical Mechanics on the Space of Probability Distributions, Neural Comp.9(2)(1997) 349–368, (arXiv:cond-mat/9601030).

The interpretation of a quantum field theory as a probability distribution on the space of field configurations so as to allow the conversion of techniques from information geometry to analogous measures of proximity between QFTs is in

• Vijay Balasubramanian, Jonathan Heckman, Alexander Maloney, Relative Entropy and Proximity of Quantum Field Theories, (arXiv:1410.6809)

A treatment of collective statistical inference as resulting in the partition function of a non-linear sigma model is in

• Jonathan Heckman, Statistical Inference and String Theory, (arXiv:1305.3621)

More in the context of quantum field theory:

• Johanna Erdmenger, Kevin T. Grosvenor, Ro Jefferson, Information geometry in quantum field theory: lessons from simple examples (arXiv:2001.02683)