nLab stochastic dependence and independence



Measure and probability theory

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In measure-theoretic probability

Let (Ω,,p)(\Omega,\mathcal{F},p) be a probability space, and let A,BA,B\in\mathcal{F} be events, i.e. measurable subsets of Ω\Omega. We say that AA and BB are independent if and only if

p(AB)=p(A)p(B), p(A\cap B) = p(A)\,p(B) ,

i.e. if the joint probability is the product of the probabilities.

More generally, if f:(Ω,)(X,𝒜)f:(\Omega,\mathcal{F})\to(X,\mathcal{A}) and g:(Ω,)(Y,)g:(\Omega,\mathcal{F})\to(Y,\mathcal{B}) are random variables or random elements, one says that ff and gg are independent if and only if all the events they induce are independent, i.e. for every A𝒜A\in\mathcal{A} and BB\in\mathcal{B},

p(f 1(A)g 1(B))=p(f 1(A))p(g 1(B)). p\big(f^{-1}(A)\cap g^{-1}(B)\big) = p\big(f^{-1}(A)\big)\,p\big(g^{-1}(B)\big) .

Equivalently, one can form the joint random variable (f,g):(Ω,)(X×Y,𝒜)(f,g):(\Omega,\mathcal{F})\to(X\times Y,\mathcal{A}\otimes\mathcal{B}) and form the joint distribution q=(f,g) *pq=(f,g)_*p on X×YX\times Y. We have that ff and gg are independent as random variables if and only if for every A𝒜A\in\mathcal{A} and BB\in\mathcal{B},

q(π 1 1(A)π 2 1(B))=q(π 1 1(A))q(π 2 1(B)). q\big(\pi_1^{-1}(A)\cap \pi_2^{-1}(B)\big) = q\big(\pi_1^{-1}(A)\big)\,q\big(\pi_2^{-1}(B)\big) .

If we denote the marginal distributions of qq by q Xq_X and q Yq_Y, the independence condition reads

q(A×B)=q X(A)q Y(B), q(A \times B) = q_X(A)\,q_Y(B) ,

meaning that the joint distribution qq is the product of its marginals.


In terms of the Giry monad


In the category of Markov kernels


In Markov categories

(For now see Markov category - Stochastic independence)

See also


  • 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)

category: probability

Last revised on February 22, 2024 at 16:11:37. See the history of this page for a list of all contributions to it.