# Causal inference from multivariate putative cause and univariate putative effect

Suppose we want to find out if observed multivariate binary random variable $\textbf{X}$ causes observed binary random variable $Y$ in presence of observed multivariate binary covariates $\textbf{Z}$. Note that $\textbf{X}, Y$, and $\textbf{Z}$ can be arbitrary. We have information such as, whether $X_i \in \textbf{X}$ causes $Y$. However, we know nothing more about those variables apart from the fact that $\textbf{X}$ and $Y$ are correlated.

A straight-forward way to approach this problem is via factorial design. That is we set $2^{|\textbf{X}|}$ treatments, and proceed with the potential outcome framework for multiple treatments. However, the number of treatments grows exponentially with the cardinality of $\textbf{X}$. Consequently stratification will also lead to treatment groups with insufficient number of samples. The method becomes practically useless for data with $n \ll 2^{|\textbf{X}|}$.

Is there a way to solve this problem computationally faster without sacrificing too much in terms of causal quantification? How can we design treatments by only knowing that $\textbf{X}$ is a multivariate binary random variable?

• Your experimental design should be feasible with the resources you have. If it's not possible, a different scientific question must be asked. – AdamO Dec 19 '17 at 15:34