nLab
Markov category

Idea

The formalism of Markov categories can be thought of as a way to express certain aspects of probability and statistics synthetically. In other words, it consists of structures and axioms which one can think of as “fundamental” in probability and statistics, which one can use to prove theorems, without having to use measure theory directly. One then proves that the usual measure-theoretic treatment is a model (or semantics) of this theory.

Intuitively, for the purposes of probability a Markov category can be seen as a category where morphisms behave like “random functions”, or “Markov kernels?” (hence the name). Canonical examples are Kleisli categories of probability monads. The formalism is however far more general.

Caveat: some of the content of this page reflects work in progress. Content may change.

Definition

A Markov category is a semicartesian symmetric monoidal category $(C,\otimes,1)$ in which every object $X$ is equipped with the structure of a commutative internal comonoid. We denote the comultiplication and counit maps by $copy: X \to X \otimes X$ and $delete: X\to 1$.

We require the following compatibility property between the copy map and the tensor product: for all objects $X$ and $Y$,

$$copy_{X\otimes Y} \; =\; (id_X\otimes b_{Y,X} \otimes id_Y) ( copy_X \otimes copy_Y ),$$

where $b$ denotes the braiding.

Note that the map $delete: X\to 1$ is uniquely determined by the fact that 1 is terminal, hence it is also natural in $X$ (see semicartesian monoidal category for more). On the other hand, the copy map is not required to be natural.

In terms of string diagrams

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Examples

• The category $\mathsf{FinStoch}$ of finite sets and stochastic matrices.
• The category $\mathsf{Stoch}$ of measurable spaces and Markov kernels?
• For any cartesian monoidal category $\mathsf{C}$ equipped with a monoidal monad $T$, the Kleisli category $Kl(T)$ is a Markov category.

See also the detailed list below.

Deterministic morphisms

A morphism $f:X\to Y$ in a Markov category is called deterministic if it commutes with the copy map,

$$copy \circ f \;=\; (f\otimes f) \circ copy .$$

A way to motivate the definition is the following. Suppose that $f$ is a “random” function between real numbers, which adds to the input the result of the roll of a die. Given a number $x$, we can roll a die, add the resulting value (say, $n$) to $x$, and then copy the result, to get $(x+n,x+n)$. Or, we could copy the value $x$, roll two dice (or roll the die twice), and add the two resulting values (say, $m$ and $n$) to the two copies of $x$, obtaining $(x+m,x+n)$. The two results are likely to differ. One can take this as a definition of randomness: it’s a process that may give a different result if you do it twice. Or equivalently, that copying the information before or after the process has taken place gives different results.

A deterministic morphism is instead one that does not exhibit this behavior, i.e. that commutes with the copy map.

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Further structures and properties

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Conditionals

Kolmogorov products

Positivity

(to be expanded)

Representable Markov categories

For now, see probability monad.

Detailed list

(…to be expanded…)

Category	Probability monad	Conditionals	Positivity	Kolmogorov products	Further references
Stoch?	Giry monad on Meas	No (Fritz'19, Example 11.3)	(…)	(…)	Fritz'19
BorelStoch?	Giry monad on Pol?	Yes (Kallenberg '17, B-M'19)	(…)	(…)	Fritz'19
FinStoch?	Not representable	Yes (Fritz'19, Example 11.6)	(…)	(…)	Fritz'19

See also

• probability monad
• probability theory
• statistics
• zero-one law?
• doctrine

References

Markov categories as defined here appear in:

• Tobias Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics, 2019. (arXiv:1908.07021)

• Tobias Fritz and Eigil Fjeldgren Rischel, The zero-one laws of Kolmogorov and Hewitt–Savage in categorical probability, 2019. (arXiv:1912.02769)

• Tobias Fritz, Tomáš Gonda, Paolo Perrone, Eigil Fjeldgren Rischel, Representable Markov categories and comparison of statistical experiments in categorical probability, arXiv:2010.07416

See also the references therein.

The first idea of defining a “category of probabilistic mappings” seems to be due to Lawvere, in

W. Lawvere, The category of probabilistic mappings, ms. 12 pages, 1962 (Lawvere Probability 1962)

(…more to come…)

Further references:

• Olaf Kallenberg, Random Measures, Theory and Applications, Springer, 2017.

• Vladimir Bogachev and Il’ya Malofeev, Kantorovich problems and conditional measures depending on a parameter. (arXiv:1904.03642)