  Developing models that are capable of answering questions of the form "How
would x change if y had been z?'" is fundamental to advancing medical image
analysis. Training causal generative models that address such counterfactual
questions, though, currently requires that all relevant variables have been
observed and that the corresponding labels are available in the training data.
However, clinical data may not have complete records for all patients and state
of the art causal generative models are unable to take full advantage of this.
We thus develop, for the first time, a semi-supervised deep causal generative
model that exploits the causal relationships between variables to maximise the
use of all available data. We explore this in the setting where each sample is
either fully labelled or fully unlabelled, as well as the more clinically
realistic case of having different labels missing for each sample. We leverage
techniques from causal inference to infer missing values and subsequently
generate realistic counterfactuals, even for samples with incomplete labels.
