A Language for Counterfactual Generative Models

We present Omega, a probabilistic programming language with support for counterfactual inference. Counterfactual inference means to observe some fact in the present, and infer what would have happened had some past intervention been taken, e.g. ``given that medication was not effective at dose x, what is the probability that it would have been effective at dose 2x?.'' We accomplish this by introducing a new operator to probabilistic programming akin to Pearl's do, define its formal semantics, provide an implementation, and demonstrate its utility through examples in a variety of simulation models.