  Causal effect estimation is important for many tasks in the natural and
social sciences. We design algorithms for the continuous partial identification
problem: bounding the effects of multivariate, continuous treatments when
unmeasured confounding makes identification impossible. Specifically, we cast
causal effects as objective functions within a constrained optimization
problem, and minimize/maximize these functions to obtain bounds. We combine
flexible learning algorithms with Monte Carlo methods to implement a family of
solutions under the name of stochastic causal programming. In particular, we
show how the generic framework can be efficiently formulated in settings where
auxiliary variables are clustered into pre-treatment and post-treatment sets,
where no fine-grained causal graph can be easily specified. In these settings,
we can avoid the need for fully specifying the distribution family of hidden
common causes. Monte Carlo computation is also much simplified, leading to
algorithms which are more computationally stable against alternatives.
