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Let $P$ be a probability distribution on a set $V$ of variables and for any $X\subseteq V$ and any possible realization $x$ of $X$ let $P_x$ be a distribution on $V\setminus X$. Let $P^*$ the collection of such distributions. Pearl (page 24 of Causality, 2009) says that a DAG $R$ is a Causal Bayesian Network compatible with $P^*$ if

  1. $P$ is Markov with respect to $R$
  2. For any $X\subseteq V$ and any realization $x$ of $X$, $P_x(v)=\prod_{i\ st\ V_i\notin X} P(v_i|pa_R(V_i))$ where $v$ is any realization of $V\setminus X$

This is actually a slight rephrasing of Pearl, in the sense that he consider interventional distributions as distributions on $V$, and asks 2. for realizations of $V$ which are consistent with $x$, in the sense that they are of the form $v=(x,v)$, up to permutation.

My question is: given a dag $R$ can we charachterise the sets of probability distributions $P^*$ for which it is a Causal Bayesian Network?

"Algorithmically" one would first pin down all the distributions which are Markov to $G$, hence those for which the pattern of conditional independencies coincides with the d-separation pattern of $G$. Then, each one should pin down uniquely a family of interventional distributions. My question is: can we have a nicer description of this set, at least in special cases (e.g. restricting to Gaussian or Bernoulli vectors) ?

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  • $\begingroup$ In point 1. you refer to $G$, but I don't think you ever introduce $G$. I assume it is another causal graph. Could you can clarify the relationship to $P$ and $R$? $\endgroup$
    – Scriddie
    Commented Mar 14, 2023 at 9:35
  • $\begingroup$ Sorry, it's just a typo. I mean $R$ there $\endgroup$ Commented Mar 15, 2023 at 8:07

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