With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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Let be a probability space, and let be events, i.e. measurable subsets of . We say that and are independent if and only if
i.e. if the joint probability is the product of the probabilities.
More generally, if and are random variables or random elements, one says that and are independent if and only if all the events they induce are independent, i.e. for every and ,
Equivalently, one can form the joint random variable and form the joint distribution on . We have that and are independent as random variables if and only if for every and ,
If we denote the marginal distributions of by and , the independence condition reads
meaning that the joint distribution is the product of its marginals.
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(For now see Markov category - Stochastic independence)
Kenta Cho, Bart Jacobs, Disintegration and Bayesian Inversion via String Diagrams, Mathematical Structures of Computer Science 29, 2019. (arXiv:1709.00322)
Tobias Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics, Advances of Mathematics 370, 2020. (arXiv:1908.07021)
Last revised on February 22, 2024 at 16:11:37. See the history of this page for a list of all contributions to it.