In mathematics, the Giry monad is a construction that assigns to a measurable space a space of probability measures over it, equipped with a canonical sigma-algebra.
[1][2][3][4][5] It is one of the main examples of a probability monad.
It is implicitly used in probability theory whenever one considers probability measures which depend measurably on a parameter (giving rise to Markov kernels), or when one has probability measures over probability measures (such as in de Finetti's theorem).
Like many iterable constructions, it has the category-theoretic structure of a monad, on the category of measurable spaces.
the set of probability measures over
to be the smallest sigma-algebra which makes the maps
is assumed equipped with the Borel sigma-algebra).
can be defined as the smallest sigma-algebra on
is part of an endofunctor on the category of measurable spaces, usually denoted again by
{\displaystyle f_{*}:(PX,{\mathcal {PF}})\to (PY,{\mathcal {PG}})}
, defined on measurable subsets
We define the probability measure
This gives a measurable, natural map
{\displaystyle {\mathcal {E}}:(PPX,{\mathcal {PPF}})\to (PX,{\mathcal {PF}})}
, and consider the probability measure
, which in this case is discrete, is given by More generally, the map
can be seen as the most general, non-parametric way to form arbitrary mixture or compound distributions.
is called the Giry monad.
, we have a bijective correspondence between measurable functions
This allows to view a Markov kernel, equivalently, as a measurably parametrized probability measure.
[10] In more detail, given a measurable function
, one can obtain the Markov kernel
Conversely, given a Markov kernel
, one can form the measurable function
From the point of view of category theory, we can interpret this correspondence as an adjunction between the category of measurable spaces and the category of Markov kernels.
, one can form the measurable space
, one can form the product measure
This gives a natural, measurable map usually denoted by
is in general not an isomorphism, since there are probability measures on
which are not product distributions, for example in case of correlation.