  Sequential experimental design to discover interventions that achieve a
desired outcome is a key problem in various domains including science,
engineering and public policy. When the space of possible interventions is
large, making an exhaustive search infeasible, experimental design strategies
are needed. In this context, encoding the causal relationships between the
variables, and thus the effect of interventions on the system, is critical for
identifying desirable interventions more efficiently. Here, we develop a causal
active learning strategy to identify interventions that are optimal, as
measured by the discrepancy between the post-interventional mean of the
distribution and a desired target mean. The approach employs a Bayesian update
for the causal model and prioritizes interventions using a carefully designed,
causally informed acquisition function. This acquisition function is evaluated
in closed form, allowing for fast optimization. The resulting algorithms are
theoretically grounded with information-theoretic bounds and provable
consistency results for linear causal models with known causal graph. We apply
our approach to both synthetic data and single-cell transcriptomic data from
Perturb-CITE-seq experiments to identify optimal perturbations that induce a
specific cell state transition. The causally informed acquisition function
generally outperforms existing criteria allowing for optimal intervention
design with fewer but carefully selected samples.
