SEMINARS
The intersection axiom of conditional independence: some new results

2019-10-02　10:30 — 12:00

Middle Lecture Room(703)

Richard Gill

University of Leiden

I will talk about some preliminary results concerning the axiomatization of conditional independence. It has been noticed that a number of properties of conditional independence between random variables bear striking resemblance to graph separation properties. Notation: suppose we have some random variables $X_1$, … , $X(n)$, then for any subset $A$ of $\{1, …, n\}$ we define $X(A)$ to be the random vector $(X(i): i \in A)$. Now we can write $A \perp B | C$ to mean that $X(A)$ is statistically independent of $X(B)$ conditional on $X(C)$. It turns out that under positivity conditions, there is a simple undirected graph $G$ on vertices $\{1, …, n\}$ such that $C$ separates $A$ from $B$ if and only if $X(A)$ is independent of $X(B)$ given $X(C)$.

Phil Dawid (1980) discovered and named what he called the (semi)-graphoid axioms of conditional independence. One of them is the “intersection axiom” corresponding to the following true fact about graph separation: $A$ is separated from $C$ by $B$ and $A$ is separated from $B$ by $C$ implies that $A$ is separated from $B$ and $C$. It corresponds to a true property of conditional probability if the random variables concerned are discrete and their joint probability mass function is strictly positive. I will discuss necessary and sufficient conditions for the intersection axiom to hold, inspired by recent work in algebraic statistics by Jonas Peters and others.

The slides of this talk can be found here.