2 added 46 characters in body

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 justonly knowing that $$\textbf{X}$$ is a multivariate binary random variable?

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 just knowing $$\textbf{X}$$?

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?

1

# 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 just knowing $$\textbf{X}$$?