Statistical manifold
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In
Examples
The family of all
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
Definition
Let X be an
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]
See also
References
- ISBN 0-412-39860-5.