  To perform counterfactual reasoning in Structural Causal Models (SCMs), one
needs to know the causal mechanisms, which provide factorizations of
conditional distributions into noise sources and deterministic functions
mapping realizations of noise to samples. Unfortunately, the causal mechanism
is not uniquely identified by data that can be gathered by observing and
interacting with the world, so there remains the question of how to choose
causal mechanisms. In recent work, Oberst & Sontag (2019) propose Gumbel-max
SCMs, which use Gumbel-max reparameterizations as the causal mechanism due to
an intuitively appealing counterfactual stability property. In this work, we
instead argue for choosing a causal mechanism that is best under a quantitative
criteria such as minimizing variance when estimating counterfactual treatment
effects. We propose a parameterized family of causal mechanisms that generalize
Gumbel-max. We show that they can be trained to minimize counterfactual effect
variance and other losses on a distribution of queries of interest, yielding
lower variance estimates of counterfactual treatment effect than fixed
alternatives, also generalizing to queries not seen at training time.
