  Maximizing a target variable as an operational objective in a structural
causal model is an important problem. Existing Causal Bayesian
Optimization~(CBO) methods either rely on hard interventions that alter the
causal structure to maximize the reward; or introduce action nodes to
endogenous variables so that the data generation mechanisms are adjusted to
achieve the objective. In this paper, a novel method is introduced to learn the
distribution of exogenous variables, which is typically ignored or marginalized
through expectation by existing methods. Exogenous distribution learning
improves the approximation accuracy of structural causal models in a surrogate
model that is usually trained with limited observational data. Moreover, the
learned exogenous distribution extends existing CBO to general causal schemes
beyond Additive Noise Models~(ANM). The recovery of exogenous variables allows
us to use a more flexible prior for noise or unobserved hidden variables. We
develop a new CBO method by leveraging the learned exogenous distribution.
Experiments on different datasets and applications show the benefits of our
proposed method.
