nLab stochastic dependence and independence

Idea

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Intuition

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In measure-theoretic probability

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

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

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

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

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

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

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

If we denote the marginal distributions of $q$ by $q_X$ and $q_Y$, the independence condition reads

$$q(A \times B) = q_X(A)\,q_Y(B) ,$$

meaning that the joint distribution $q$ is the product of its marginals.

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In terms of the Giry monad

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In the category of Markov kernels

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In Markov categories

(For now see Markov category - Stochastic independence)

See also

• probability theory, categorical probability
• joint and marginal probability
• Kolmogorov extension theorem
• probability monad, Giry monad
• Markov category, Markov kernel

References

• 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