In
mathematics, a statistical manifold is a
Riemannian manifold, each of whose points is a
probability distribution. Statistical manifolds provide a setting for the field of
information geometry. The
Fisher information metric provides a
metric on these manifolds. Following this definition, the
log-likelihood function is a
differentiable map and the
score is an
inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society.
** Inclusion (disability rights), promotion of people with disabiliti ...
.
Examples
The family of all
normal distributions can be thought of as a 2-dimensional parametric space parametrized by the
expected value ''μ'' and the
variance ''σ''
2 ≥ 0. Equipped with the
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
given by the
Fisher information
In mathematical statistics, the Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable ''X'' carries about an unknown parameter ''θ'' of a distribution that model ...
matrix, it is a statistical manifold with a geometry modeled on
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
. A way of picturing the manifold is done by inferring the parametric equations via the Fisher Information rather than starting from the likelihood-function.
A simple example of a statistical manifold, taken from physics, would be the
canonical ensemble: it is a one-dimensional manifold, with the
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer.
Thermometers are calibrated in various Conversion of units of temperature, temp ...
''T'' serving as the coordinate on the manifold. For any fixed temperature ''T'', one has a probability space: so, for a gas of atoms, it would be the probability distribution of the velocities of the atoms. As one varies the temperature ''T'', the probability distribution varies.
Another simple example, taken from medicine, would be the probability distribution of patient outcomes, in response to the quantity of medicine administered. That is, for a fixed dose, some patients improve, and some do not: this is the base probability space. If the dosage is varied, then the probability of outcomes changes. Thus, the dosage is the coordinate on the manifold. To be a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, one would have to measure outcomes in response to arbitrarily small changes in dosage; this is not a practically realizable example, unless one has a pre-existing mathematical model of dose-response where the dose can be arbitrarily varied.
Definition
Let ''X'' be an
orientable manifold, and let
be a
measure on ''X''. Equivalently, let
be a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
on
, with
sigma algebra and probability
.
The statistical manifold ''S''(''X'') of ''X'' is defined as the space of all measures
on ''X'' (with the sigma-algebra
held fixed). Note that this space is infinite-dimensional; it is commonly taken to be a
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
. The points of ''S''(''X'') are measures.
Rather than dealing with an infinite-dimensional space ''S''(''X''), it is common to work with a finite-dimensional
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
, defined by considering a set of
probability distributions parameterized by some smooth, continuously-varying parameter
. That is, one considers only those measures that are selected by the parameter. If the parameter
is ''n''-dimensional, then, in general, the submanifold will be as well. All finite-dimensional statistical manifolds can be understood in this way.
References
Manifolds
Information theory
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