Title: Visible and Invisible: Causal Variable Learning and its Application in a Cancer Study. Abstract: Causal visual discovery is a fundamental yet challenging problem in many research fields. Given visual data and the outcome of interest, the goal is to infer the cause-effect relation. Aside from rich visual ('visible') variables, oftentimes, the outcome is also determined by 'invisible' variables, i.e. the variables from non-visual modalities that do not have visual counterparts. This $(\textbf{visible}, \textbf{invisible})$ combination is particularly common in the clinical domain.

Built upon the promising invariant causal prediction (ICP) framework, we propose a novel $\varepsilon$-ICP algorithm to resolve the (visible, invisible) setting. To efficiently discover $\varepsilon$-plausible causal variables and to estimate the cause-effect relation, the $\varepsilon$-ICP is learned under a min-min optimisation scheme. Driven by the need for clinical reliability and interpretability, the $\varepsilon$-ICP is implemented with a typed neural-symbolic functional language. With the built-in program synthesis method, we can synthesize a type-safe program that is comprehensible to the clinical experts.

For concept validation of the $\varepsilon$-ICP, we carefully design a series of synthetic experiments on the type of visual-perception tasks that are encountered in daily life. To further substantiate the proposed method, we demonstrate the application of $\varepsilon$-ICP on a real-world cancer study dataset, Swiss CRC. This population-based cancer study has spanned over two decades, including 25$k$ fully annotated tissue micro-array (TMA) images with at least $3k \times 3k$ resolution and a broad spectrum of clinical meta data for 533 patients. Both the synthetic and clinical experiments demonstrate the advantages of $\varepsilon$-ICP over the state-of-the-art methods. Finally, we discuss the limitations and challenges to be addressed in the future.
