Introduction
First of all, this article is best viewed in Chromium, Google Chrome or Chrome based browsers. Some characters are not rendered correctly in Firefox. In this article, an important fact related with the proof of a theorem is proven. The theorem is about the existence of an admissible sequence within the context of the sequential backdoor criterion. The theorem states the following:
If there exists an admissible sequence $Z_{1}^{*}, \ldots , Z_{n}^{*}$, then for every minimally admissible subsequence $Z_{1}, \ldots , Z_{k-1}$ of covariates, there is an admissible set $Z_{k}$.
The important fact related with this theorem is as follows:
The covariates in the minimally admissible sequence $Z_{1}, \ldots , Z_{k-1}, Z_{k}$ cannot contain a nonmember of $A_{k}$ where $A_{k}$ is the ancestral set of $\left(X_{k}, Y\right)$ in $G_{k}$ which is $G_{\underline{X}_{k}, \overline{X}_{k+1}, \ldots , \overline{X}_{n}}$.
This important fact was already proven in the article [1]. However, in my opinion, this proof is not clear enough and redundantly long without explanations. That’s why I want to elaborate on it.
The Proof
If $Z_{1}, \ldots , Z_{k-1}, Z_{k}$ is a minimally admissible sequence then $Z_{k} \subseteq N_{k}$ and \begin{equation} \left(Y \perp \!\!\! \perp X_{k} | X_{1}, \ldots , X_{k-1}, Z_{1}, \ldots , Z_{k-1}, Z_{k}\right ) \text{ in } G_{k} \label{admissibility2} \end{equation} where \begin{equation} G_{k} = G_{\underline{X}_{k}, \overline{X}_{k+1}, \ldots , \overline{X}_{n}} \end{equation} and $N_{k}$ is the set of covariates which are nondescendants of $X_{k}$. The minimality constraint requires the covariates $Z_{1}, \ldots , Z_{k-1}, Z_{k}$ to be minimal. The fact that the covariates in the minimally admissible sequence $Z_{1}, \ldots , Z_{k-1}, Z_{k}$ cannot contain a nonmember of $A_{k}$ in $G_{k}$ is to be proven using contradiction. As a contradiction, it is assumed that $Z_{i}$ $\left(1 \leq i \leq k\right)$ contains nonmembers of $A_{k}$. Let the nonmembers of $A_{k}$ be removed from $Z_{i}$. Removing the nonmembers of $A_{k}$ from $Z_{i}$ must not break the conditional independence in the equation \eqref{admissibility2}. The reason is the second equation \eqref{lemma12} of the lemma 1 in [1]. Lemma 1 is given as follows:
For any DAG $G$ and any two disjoint subsets of nodes $X$ and $Y$, let the ancestor-set of $\left(X,Y\right)$, denoted $A\left(X,Y\right)$, be the set of nodes which have a descendant in either $X$ or $Y$. The following two separation conditions hold for any sets of nodes $W$ and $Z$: \begin{equation} \left(Y \perp \!\!\! \perp X | Z, W \cap A\left(X,Y\right)\right)_{G} \text{ whenever } \left(Y \perp \!\!\! \perp X|Z\right)_{G} \label{lemma11} \end{equation} \begin{equation} \left(Y \perp \!\!\! \perp X|W\cap A\left(X,Y\right) \right)_{G} \text{ whenever } \left(Y \perp \!\!\! \perp X|W\right)_{G} \label{lemma12} \end{equation}
The second equation \eqref{lemma12} of this lemma states that removing nonmembers of the ancestors from the conditioning set does not harm conditional independence. The proof of this lemma is elaborated on in [2]. Let the form of $Z_{i}$ with nonmembers of $A_{k}$ removed be denoted by $Z_{i}^{\prime}$. Then, the following conditional dependence is known to hold due to \eqref{lemma12}: \begin{equation} \left(Y \perp \!\!\! \perp X_{k} | X_{1}, \ldots , X_{k-1}, Z_{1}, \ldots , Z_{i}^{\prime}, \ldots , Z_{k-1}, Z_{k}\right ) \text{ in } G_{k} \label{admissibility3} \end{equation} In the equation \eqref{admissibility3}, $Z_{i}$ in the conditioning set can be seen to have been replaced with $Z_{i}^{\prime}$ which is smaller than $Z_{i}$. Then, $Z_{i}$ is not minimal. This is a contradiction. This contradiction has been caused by the assumption that $Z_{i}$ contains nonmembers of $A_{k}$. So, it has been proven by contradiction that $Z_{i}$ $\left(1\leq i \leq k\right)$ does not contain nonmembers of $A_{k}$.
Conclusion
The proof of the fact that none of minimally admissible subsequences can contain nonmembers of the ancestral set has been elaborated on.
References
[1] Judea Pearl and James Robins. Probabilistic evaluation of sequential plans from causal models with hidden variables. In Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, pages 444-453, 1995.