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statistical manifold
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.
==Examples==
The family of all normal distributions, parametrized by the expected value ''μ'' and the variance ''σ''2 ≥ 0, with the Riemannian metric given by the Fisher information matrix, is a statistical manifold. Its geometry is modeled on hyperbolic space.
A simple example of a statistical manifold, taken from physics, would be the canonical ensemble: it is a one-dimensional manifold, with the temperature ''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, 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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