  We propose functional causal Bayesian optimization (fCBO), a method for
finding interventions that optimize a target variable in a known causal graph.
fCBO extends the CBO family of methods to enable functional interventions,
which set a variable to be a deterministic function of other variables in the
graph. fCBO models the unknown objectives with Gaussian processes whose inputs
are defined in a reproducing kernel Hilbert space, thus allowing to compute
distances among vector-valued functions. In turn, this enables to sequentially
select functions to explore by maximizing an expected improvement acquisition
functional while keeping the typical computational tractability of standard BO
settings. We introduce graphical criteria that establish when considering
functional interventions allows attaining better target effects, and conditions
under which selected interventions are also optimal for conditional target
effects. We demonstrate the benefits of the method in a synthetic and in a
real-world causal graph.
