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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.00847 |
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| _version_ | 1866908415022858240 |
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| author | Mayer, Matthias Georg |
| author_facet | Mayer, Matthias Georg |
| contents | 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 $(Ω, \mathcal{A}, \mathbb{P})$. Let $X$, $Y$, and $Z$ be arbitrary $σ(U)$-measurable random elements. We characterize all independences $X \perp Y \mid Z$ implied by the independence of $U$ and call these independences \textit{structural}. Formally, these are the independences which hold in all probability measures $P$ that render $U$ independent and are absolutely continuous with respect to $\mathbb{P}$; i.e., for all such $P$, it must hold that $X \perp_P Y \mid Z$.
We introduce the history $\mathcal{H}(X \mid Z) : Ω\to \mathcal{P}(I)$, 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 $\mathcal{H}(X \mid Z) \cap \mathcal{H}(Y \mid Z) = \emptyset$ almost surely with respect to $\mathbb{P}$.
Finally, we apply this $d$-separation-like criterion in structural causal models to discover a causal direction in a toy setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00847 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Theory of Structural Independence Mayer, Matthias Georg Probability 60A99 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 $(Ω, \mathcal{A}, \mathbb{P})$. Let $X$, $Y$, and $Z$ be arbitrary $σ(U)$-measurable random elements. We characterize all independences $X \perp Y \mid Z$ implied by the independence of $U$ and call these independences \textit{structural}. Formally, these are the independences which hold in all probability measures $P$ that render $U$ independent and are absolutely continuous with respect to $\mathbb{P}$; i.e., for all such $P$, it must hold that $X \perp_P Y \mid Z$. We introduce the history $\mathcal{H}(X \mid Z) : Ω\to \mathcal{P}(I)$, 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 $\mathcal{H}(X \mid Z) \cap \mathcal{H}(Y \mid Z) = \emptyset$ almost surely with respect to $\mathbb{P}$. Finally, we apply this $d$-separation-like criterion in structural causal models to discover a causal direction in a toy setting. |
| title | A Theory of Structural Independence |
| topic | Probability 60A99 |
| url | https://arxiv.org/abs/2412.00847 |