Transparent Parametrizations of Models for Potential Outcomes

We consider causal models involving three binary variables: a randomized assignment Z, an exposure measure X, and a final response Y . We focus particular attention on the situation in which there may be confounding of X and Y , while at the same time measures of the e⇤ect of X on Y are of primary interest. In the case where Z has no e⇤ect on Y , other than through Z, this is the instrumental variable model. Many causal quantities of interest are only partially identified. We first show via an example that the resulting posteriors may be highly sensitive to the specification of the prior distribution over compliance types. To address this, we present several novel “transparent” re-parametrizations of the likelihood that separate the identified and nonidentified parts of the parameter. In addition, we develop parametrizations that are robust to model mis-specification under the “intent-to-treat” null hypothesis that Z and Y are independent.