A Theory of Structural Independence

[Matthias G. Mayer]

Link post

The linked paper introduces the key concept of factored spaced models /​ finite factored sets, structural independence, in a fully general setting using families of random elements. The key contribution is a general definition of the history object and a theorem that the history fully characterizes the semantic implications of the assumption that a family of random elements is independent. This is analogous to how d-separation precisely characterizes which nodal variables are independent given some nodal variables in any probability distribution which fulfills the markov property on the graph.

Abstract: Structural independence is the (conditional) independence that arises from the structure rather than the precise numerical values of a distribution. We develop this concept and relate it to d-separation and structural causal models.

Formally, let $U=(U_i{)}_{i\in I}$ be an independent family of random elements on a probability space $(\Omega ,A,P)$. Let $X$, $Y$, and $Z$ be arbitrary $\sigma \left(U\right)$-measurable random elements. We characterize all independences $X\perp \perp Y\mid Z$ implied by the independence of $U$ and call these independences structural. Formally, these are the independences which hold in all probability measures $P$ that render $U$ independent and are absolutely continuous with respect to $P$, i.e., for all such $P$, it must hold that $X{\perp }_{P}Y\mid Z$.

We introduce the history $\mathrm{history}(X\mid Z):\Omega \to P\left(I\right)$, a combinatorial object that measures the dependence of $X$ on $U_i$ for each $i\in I$ given $Z$. The independence of $X$ and $Y$ given $Z$ is implied by the independence of $U$ if and only if $\mathrm{history}(X\mid Z)\cap \mathrm{history}(Y\mid Z)=\varnothing$ almost surely with respect to $P$.

Finally, we apply this d-separation-like criterion in structural causal models to discover a causal direction in a toy setting.

[Alexander Gietelink Oldenziel]

What is meant by ‘causal direction’ ? Where in the paper is this referring to?

[Matthias G. Mayer]

By causal direction (X causes Y) I mean that in the induced graph of the structural causal model, X appears before Y. The graph induced by a structural causal model has an edge from X to Y if there is a structural equation Y = F(U_i, X, … ).

It refers to Example 9.3 and the associated lemmas.

This example is a general rephrasing of Example 1 of finite factored sets in terms of random variables and structural causal models.

Example 9.3

We observe two nonconstant, (real valued) random variables $V_1$ and $V_2$. We assume that they are part of an unobserved true underlying causal model (𝑈, 𝑉 , 𝐹, ℙ), where 𝑉 = $(V_i{)}_{i\in I}$ and {1, 2} ⊆ 𝐼. We only observe the distribution of 𝑉 . Let $Z≔V_1+V_2$. Suppose that $V_1{⫫}_{P}Z$. Furthermore, we assume that ℙ is in ‘general position’. This means that this independence is not due to the specific choice of ℙ. Formally, $V_1{⫫}_{P}Z$ for all causal models (𝑈, 𝑉 , 𝐹, 𝑃) s.t. 𝑃 ∼ ℙ.

It says that any structural causal model that models the independence of $V_1$ and Z structurally (it is not a coincidence of precise numerical values of ℙ), then in this structural causal model has $V_1\to V_2$, i.e. in the graph induced by the causal model through it’s equation, $V_1$ always is an ancestor of $V_2$ .