Construction
The Giry monad, like every monad, consists of three structures: [6] [7] [8]
- A functorial assignment, which in this case assigns to a measurable space
a space of probability measures
over it; - A natural map
called the unit, which in this case assigns to each element of a space the Dirac measure over it; - A natural map
called the multiplication, which in this case assigns to each probability measure over probability measures its expected value.
The space of probability measures
Let
be a measurable space. Denote by
the set of probability measures over
. We equip the set
with a sigma-algebra as follows. First of all, for every measurable set
, define the map
by
. We then define the sigma algebra
on
to be the smallest sigma-algebra which makes the maps
measurable, for all
(where
is assumed equipped with the Borel sigma-algebra). [6]
Equivalently,
can be defined as the smallest sigma-algebra on
which makes the maps

measurable for all bounded measurable
. [9]
The assignment
is part of an endofunctor on the category of measurable spaces, usually denoted again by
. Its action on morphisms, i.e. on measurable maps, is via the pushforward of measures. Namely, given a measurable map
, one assigns to
the map
defined by

for all
and all measurable sets
. [6]
The Dirac delta map
Given a measurable space
, the map
maps an element
to the Dirac measure
, defined on measurable subsets
by [6]

The expectation map
Let
, i.e. a probability measure over the probability measures over
. We define the probability measure
by

for all measurable
. This gives a measurable, natural map
. [6]
Example: mixture distributions
A mixture distribution, or more generally a compound distribution, can be seen as an application of the map
. Let's see this for the case of a finite mixture. Let
be probability measures on
, and consider the probability measure
given by the mixture

for all measurable
, for some weights
satisfying
. We can view the mixture
as the average
, where the measure on measures
, 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.
The triple
is called the Giry monad. [1] [2] [3] [4] [5]
Relationship with Markov kernels
One of the properties of the sigma-algebra
is that given measurable spaces
and
, we have a bijective correspondence between measurable functions
and Markov kernels
. 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
as follows,

for every
and every measurable
(note that
is a probability measure). Conversely, given a Markov kernel
, one can form the measurable function
mapping
to the probability measure
defined by

for every measurable
. The two assignments are mutually inverse.
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. In particular, the category of Markov kernels can be seen as the Kleisli category of the Giry monad. [3] [4] [5]
Product distributions
Given measurable spaces
and
, one can form the measurable space
with the product sigma-algebra, which is the product in the category of measurable spaces. Given probability measures
and
, one can form the product measure
on
. This gives a natural, measurable map

usually denoted by
or by
. [4]
The map
is in general not an isomorphism, since there are probability measures on
which are not product distributions, for example in case of correlation. However, the maps
and the isomorphism
make the Giry monad a monoidal monad, and so in particular a commutative strong monad. [4]
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