  We consider the task of causal imputation, where we aim to predict the
outcomes of some set of actions across a wide range of possible contexts. As a
running example, we consider predicting how different drugs affect cells from
different cell types. We study the index-only setting, where the actions and
contexts are categorical variables with a finite number of possible values.
Even in this simple setting, a practical challenge arises, since often only a
small subset of possible action-context pairs have been studied. Thus, models
must extrapolate to novel action-context pairs, which can be framed as a form
of matrix completion with rows indexed by actions, columns indexed by contexts,
and matrix entries corresponding to outcomes. We introduce a novel SCM-based
model class, where the outcome is expressed as a counterfactual, actions are
expressed as interventions on an instrumental variable, and contexts are
defined based on the initial state of the system. We show that, under a
linearity assumption, this setup induces a latent factor model over the matrix
of outcomes, with an additional fixed effect term. To perform causal prediction
based on this model class, we introduce simple extension to the Synthetic
Interventions estimator (Agarwal et al., 2020). We evaluate several matrix
completion approaches on the PRISM drug repurposing dataset, showing that our
method outperforms all other considered matrix completion approaches.
