## Causal Inference

**(supported in part by Frymoyer Chair in IST held by Vasant Honavar at Penn State University)**

Elicitation of a causal effect from observations and experiments is central to scientific discovery, or more generally, rational approaches to understanding and interacting with the world around us. Judea Pearl introduced causal diagrams provide a formal representation for combining data with causal information and do-calculus to provide a sound and complete inferential machinery for causal inference.

The practical need to transfer causal effects elicited in one domain (setting, environment, population) e.g., a controlled laboratory setting, to a different setting presents us with the problem of transporting causal information from a source environment to a possibly different target environment. For example, one might want to know if causal relation between teaching strategies and student learning obtained by through a randomized trial in a public school in Chicago can be transported to a public school in Minneapolis that has an admittedly different population of students. Our recent work has examined m-transportability, a generalization of transportability, which offers a license to use causal information elicited from experiments and observations in m (where m is greater than or equal to 1) source environments to estimate a causal effect in a given target environment. We have provided a novel characterization of m-transportability that directly exploits the completeness of do-calculus to obtain the necessary and sufficient conditions for m-transportability. We have designed an algorithm for deciding m-transportability that determines whether a causal relation is m-transportable; and if it is, produces a transport formula, that is, a recipe for estimating the desired causal effect by combining experimental information from m source environments with observational information from the target environment.

We have introduced z-transportability, the problem of estimating the causal effect of a set of variables X on another set of variables Y in a target domain from experiments on any subset of controllable variables Z where Z is an arbitrary subset of observable variables V in a source domain. Z-Transportability generalizes z-identifiability, the problem of estimating in a given domain the causal effect of X on Y from surrogate experiments on a set of variables Z such that Z is disjoint from X. z-Transportability also generalizes transportability which requires that the causal effect of X on Y in the target domain be estimable from experiments on any subset of all observable variables in the source domain. We have generalized z-identifiability to allow cases where Z is not necessarily disjoint from X. We have established a necessary and sufficient condition for z-transportability in terms of generalized z-identifiability and transportability. We have provided a sound and complete algorithm that determines whether a causal effect is z-transportable; and if it is, produces a transport formula, that is, a recipe for estimating the causal effect of X on Y in the target domain using information elicited from the results of experimental manipulations of Z in the source domain and observational data from the target domain. Our results also show that do-calculus is complete for z-transportability.

We have recently introduced mz-transportability, the problem of inferring a causal effect of treatment variables on observables in a target domain (environment, experimental setting) by combining data from experiments on simultaneously controllable subsets of variables (together with observations) from multiple domains (including the target domain). We have provided an efficient and complete algorithm that determines if a causal effect is mz-transportable, and if so, outputs a transport formula for estimating the causal effect. These results set the stage for considering more general forms of meta-identifiability by allowing a fully arbitrary information set and for proving the completeness of do-calculus in such settings.