I find the $d$-separation criterion (see, e.g., [Theorem 2 here](https://ftp.cs.ucla.edu/pub/stat_ser/r130-reprint.pdf); note however the preceding definition, which basically means we are treating discrete random variables) a really useful sufficient criterion for conditional independence of random variables. However, I am struggling to find a general version for arbitrary random variables, say with values in a Polish space (where conditional independence is defined via the respective disintegrations).


**My question:** Is there a version of $d$-separation that treats arbitrary Polish spaces valued random variables, or at least arbitrary $\mathbb{R}^d$ valued random variables?


*To make the post self-contained:* Let $\pi \in \mathcal{P}(S\_1\times S\_2 \times \dots \times S\_N)$ be a Borel probability measure on the polish space $S\_1 \times ... \times S\_N$. Assume there is a directed acyclic graph with nodes $\{1, 2, \dots, N\}$, where without loss of generality the nodes are increasing in topological order (i.e., there is no edge pointing from $k$ to $l$ for $k > l$). Denote by $pa(k)$ the set of parents of a node $k$. Assume $\pi$ has a regular disintegration of the form
$$
\pi = \pi\_{1} \otimes \pi\_{2, pa(2)} \otimes \pi\_{3, pa(3)} \otimes \dots \otimes \pi\_{N, pa(N)},
$$
where $\pi\_{k, pa(k)}$ is a stochastic kernel mapping from $(\otimes\_{j \in pa(k)} S\_j)$ to $\mathcal{P}(S\_k)$.
Note that this disintegration naturally encodes certain conditional independencies. Indeed, if $(X\_1, X\_2, \dots, X\_N) \sim \pi$ is a random variable with distribution $\pi$, the given disintegration means that $X\_k$ is conditionally independent of $\{X\_j : j \in \{1, ..., k-1\}\backslash pa(k)\}$ given $\{X\_j : j \in pa(k)\}$.


On the other hand, this disintegration also implies many other conditional independences between the variables, and $d$-separation is a criterion to identify all of them (atleast for discrete spaces). The definition of $d$-separation is as follows: Take disjoint $A, B, C \subset \{1, \dots, N\}$. We say that $C$ $d$-separates $A$ and $B$, if there are no paths from a node $i \in A$ to a node $j \in B$ which is not blocked by a node in $C$. A path between nodes is blocked if either of the following criteria is satisfied:


1. there is a chain element $a \rightarrow b \leftarrow c$ contained in the path such that $b \not \in C$ and none of the descendants of $b$ is in $C$.
2. there is a chain element $a \rightarrow b \rightarrow c$ or $a \leftarrow b \rightarrow c$ contained in the path such that $b \in C$.


Then the fundamental result we want is that if $C$ $d$-separates $A$ and $B$, then $\{X\_j : j\in A\}$ and $\{X\_j : j \in B\}$ are conditionally independent given $\{X\_j : j \in C\}$ (all under $\pi$, of course).



General version of $d$-separation
Yes, there are more general versions of the $d$-separation criterion, in particular also for standard Borel spaces (i.e. Polish spaces).


For completeness, the classical version of $d$-separation (i.e. on discrete random variables) has been generalized several times to the best of my knowledge.


### 1. $d$-separation for absolute continuous probability distributions


Lauritzen et al. (see [Independence Properties of Directed Markov Fields](https://onlinelibrary.wiley.com/doi/epdf/10.1002/net.3230200503)) showed that the $d$-separation criterion holds for absolute continuous probability distributions with respect to a product measure. More specifically, let $\mathcal{H}\_1, \ldots, \mathcal{H}\_n$ be measurable spaces and $P$ a probability distribution on $\prod\_{i=1}^{n} \mathcal{H}\_i$ such that
$$P(X\_1 \in E\_1, \ldots, X\_n \in E\_n) = \int\_{x\_1 \in E\_1} \cdots \int\_{x\_n \in E\_n} f(x\_1,\ldots, x\_n) \ \mu\_1(\mathrm{d} x\_1) \otimes \cdots \otimes \mu\_n(\mathrm{d} x\_n)$$
where $E\_i$ is measurable, $f$ being a density and $\mu\_i$ being a measure on $\mathcal{H}\_i$. In this situation, a distribution $P$ is called **compatible** with a DAG $G$ (with vertices $\{1, \ldots, n\}$) if the density $f$ factorizes as
$$f(x\_1, \ldots, x\_n) = \prod\_{i=1}^{n} f(x\_i | x\_{\textsf{pa}(i)}) \quad \mu\textrm{-a.e.}$$
where $\textsf{pa}(i)$ is the set of parents of vertex $i$ and $f(x\_i | x\_{\textsf{pa}(i)})$ is the conditional density of $X\_i$ given its parents.


Theorem 1 in the reference shows that the following two statements are equivalent


* $P$ is compatible with $G$
* $P$ satisfies the global Markov property, i.e. for every disjoint triple of subsets $A,B,C \subseteq \{1, \ldots, n\}$ such that $C$ d-separates $A$ and $B$ in $G$, we have that
$$f(x\_A, x\_B, x\_C) = f(x\_A | x\_C) \cdot f(x\_B | x\_C) \cdot f(x\_C) \quad \mu\textrm{-a.e.},$$
i.e. $\{X\_i: i\in A\}$ is cond. independent of $\{X\_j: j \in B\}$ given $\{X\_k: k\in C\}$.


### 2. $d$-separation for standard Borel spaces (i.e. Polish spaces)


A version of the $d$-separation criterion (with several other equivalent conditions) for standard Borel spaces has been proven in by P. Forré and J. Mooij ([Markov Properties for Graphical Models with Cycles and Latent Variables](https://arxiv.org/abs/1710.08775)). Your desired version is stated in Theorem 3.2.1. ($3. \Longleftrightarrow 6.$)


In this setting a probability distribution on standard Borel spaces $\prod\_{i=1}^n \mathcal{H}\_i$ given by
$$P(X\_1 \in E\_1, \ldots, X\_n \in E\_n) = \int\_{x\_1 \in E\_1} \cdots \int\_{x\_n \in E\_n} P(\mathrm{d} x\_1, \ldots, \mathrm{d} x\_n)$$
with $E\_{i} \in \mathcal{H}\_i$ is **compatible** with a DAG $G$ if
$$P(\mathrm{d} x\_1, \ldots, \mathrm{d} x\_n) = \prod\_{i = 1}^n P(\mathrm{d} x\_i | x\_{\textsf{pa}(i)})$$
where $P(\mathrm{d} x\_i | x\_{\textsf{pa}(i)})$ is the regular conditional probability distribution of $X\_i$ given its parents in $G$.
In Theorem 3.2.1. ($3. \Longleftrightarrow 6.$) it is proven that the following two statements are equivalent:


* $P$ is compatible with $G$
* $P$ satisfies the global Markov property, i.e. for every disjoint triple of subsets $A, B, C \subseteq \{1,\ldots, n\}$ such that $A$ is $d$-separated from $B$ by $C$, we have that $\{X\_i: i\in A\}$ is cond. independent of $\{X\_j: j \in B\}$ given $\{X\_k: k\in C\}$ where conditional independence is defined via regular conditional probabilities.


### 3. $d$-separation on Markov categories


As already mentioned in the question, conditional independence is defined via regular conditional probability distributions in all examples considered.


Recently, Tobias Fritz and me showed the $d$-separation criterion in the more abstract setting of Markov categories (see [The $d$-Separation Criterion in Categorical Probability](https://arxiv.org/abs/2207.05740)). There we have shown that whenever you can do an operation that behaves like Bayesian disintegration (i.e. whenever conditionals exist), the $d$-separation criterion applies.


This result includes the $d$-separation criterion on standard Borel spaces and finite probability distributions as specific instances. But it also includes non-probabilistic settings, where $d$-separation applies.


For example, it implies that the $d$-separation criterion also applies to finite possibilistic networks, i.e. networks where a conditional $f(x | a)$ specifies whether $x \in X$ is possible or impossible given $a \in A$ by setting $f(x | a) = 1$ or $f(x|a) = 0$.


