category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
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.
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 $Y$,
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.
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See also the detailed list below.
A morphism $f:X\to Y$ in a Markov category is called deterministic if it commutes with the copy map,
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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(to be expanded)
For now, see probability monad.
(…to be expanded…)
Category | Probability monad | Conditionals | Positivity | Kolmogorov products | Further references |
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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 |
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)
Tobias Fritz, Tomáš Gonda?, Paolo Perrone, De Finetti’s theorem in categorical probability. Journal of Stochastic Analysis, 2021. (arXiv:2105.02639)
Tobias Fritz, Wendong Liang?, Free gs-monoidal categories and free Markov categories, 2022. (arXiv:2204.02284)
Sean Moss?, Paolo Perrone, Probability monads with submonads of deterministic states, LICS 2022. (arXiv:2204.07003)
Paolo Perrone, Markov Categories and Entropy (arXiv:2212.11719)
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)
Last revised on December 23, 2022 at 14:44:38. See the history of this page for a list of all contributions to it.