Statistical manifolds provide a setting for the field of information geometry.
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 σ2 ≥ 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.
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.
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.
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.
The statistical manifold S(X) of X is defined as the space of all 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