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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.^{[1]}

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 given by the Fisher information matrix, it is a statistical manifold with a geometry modeled on hyperbolic space. 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 *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.

Let *X* be an orientable manifold, and let be a measure on *X*. Equivalently, let be a probability space 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. 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, 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.^{[clarification needed]}