  In causal inference, it is common to estimate the causal effect of a single
treatment variable on an outcome. However, practitioners may also be interested
in the effect of simultaneous interventions on multiple covariates of a fixed
target variable. We propose a novel method that allows to estimate the effect
of joint interventions using data from different experiments in which only very
few variables are manipulated. If there is only little randomized data or no
randomized data at all, one can use observational data sets if certain parental
sets are known or instrumental variables are available. If the joint causal
effect is linear, the proposed method can be used for estimation and inference
of joint causal effects, and we characterize conditions for identifiability. In
the overidentified case, we indicate how to leverage all the available causal
information across multiple data sets to efficiently estimate the causal
effects. If the dimension of the covariate vector is large, we may only have a
few samples in each data set. Under a sparsity assumption, we derive an
estimator of the causal effects in this high-dimensional scenario. In addition,
we show how to deal with the case where a lack of experimental constraints
prevents direct estimation of the causal effects. When the joint causal effects
are non-linear, we characterize conditions under which identifiability holds,
and propose a non-linear causal aggregation methodology for experimental data
sets similar to the gradient boosting algorithm where in each iteration we
combine weak learners trained on different datasets using only unconfounded
samples. We demonstrate the effectiveness of the proposed method on simulated
and semi-synthetic data.
