nLab stochastic map




In probability theory, a stochastic map between measurable spaces is a (linear) map of their measures which preserves the probability measures among them.

For finite spaces — where a measure is just a tuple (vector) of non-negative real numbers and a probability measure is such a tuple whose sum is 1 – a stochastic map is given by a matrix all whose columns sum to 11, called a stochastic matrix.

If, on compact measureable spaces, a stochastic map also preserves the uniform distribution, then it is called a doubly stochastic map.

For the case over finite spaces this means that the stochastic matrix has not just columns but also rows which sum to unity, then called a doubly stochastic matrix.

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Created on September 17, 2023 at 14:27:54. See the history of this page for a list of all contributions to it.