# How can I compute the set of interventional probability distributions compatible with a DAG?

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) ?

• 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 $R$? Commented Mar 14, 2023 at 9:35
• Sorry, it's just a typo. I mean $R$ there Commented Mar 15, 2023 at 8:07