Leonard Schulman

Topic 1: Algorithms and Information Theory for Causal Inference

Can we draw rational conclusions from data in absence of well-controlled scientific experiments that differentiate causal from correlative relationships? This question is of profound significance for society. There are many reasons that we must often make do with passive observation, including: ethical considerations, especially with regard to human subjects; governance constraints; and uniqueness of the system and the inability to re-run history, as e.g., regarding climate. Conceptually, the question raises the exciting prospect of extending the scientific method, rigorously, beyond the traditional framework of controlled or natural experiments.

Our working framework is "structured causal models", developed in the 1990s. In this approach, interactions among system variables can be represented by a directed graph; remarkably, certain graphs allow for statistically sound causal inference from passive data. Despite abundant work, this approach has so far had limited effect on practice. I argue that this is largely due to fragility of the existing theory. First, few graphs, generally only very sparse ones, allow causal identifiability. Second, the requirement regarding absence of an edge is absolute: even a slight interaction effect requires it be retained in full.

Representative goals include: (i) Developing identifiability theorems which relax the model verification requirements in exchange for "error bars" on the identified causal effect. (ii) Developing algorithms which, for suitable graphs, require as input only the probabilities of relatively likely events.

Topic 2: Coding Theory and High-Dimensional Combinatorics

One of the key tools for the interactive coding theorem are tree codes, which we know to exist, but do not know how to construct, despite many efforts. I have made a speculative construction but it depends upon a conjecture in analytic number theory that is currently out of reach. I plan to study variations which may circumvent this obstacle.

Relatedly, I am interested in "analog error-correcting codes": linear subspaces based on the Dvoretzky theorem on Euclidean Sections, a result in its essence similar to the Shannon coding theorem. I wish to develop explicit constructions of these objects (only randomized constructions are known), which would enable a power-limited analogue of the interactive coding theorem. This is closely related to problems in signal processing and control theory.