Conference

When some variables in a directed acyclic graph (DAG) are hidden, a notoriously complicated set of constraints on the distribution of observed variables is implied. In this talk, we present inequality constraints implied by graphical criteria in hidden variable DAGs. The constraints can intuitively be understood to follow from the fact that the capacity of variables along a causal pathway to convey information is restricted by their entropy. For DAGs that exhibit e-separation relations, we present entropic inequality constraints and we show how they can be used to learn about the true causal model from an observed data distribution (arXiv:2107.07087).

"If one only performs experiments involving passive observations, in general there are multiple causal structures that can explain the same set of distributions over the observed variables. In this case, we say that these causal structures are observationally equivalent. In this work, we explore all the known techniques for proving observational equivalence or inequivalence, as well as some original ones. Even if the existing rules are not enough to achieve the full classification of the causal structures with four observed variables, our results get close to such classification and show that admitting inequality constraints is a generic feature among structures with four observed variables."

Quantum measurements have been a central topic of research in quantum theory for many years. In the context of causal structures and communication over networks, we are often particularly interested in local measurements of subsystems of a multi-partite system and classical processing of their inputs and outcomes. Formally, this processing can often be described by means of maps that are known as wirings. These wirings are furthermore interesting for the analysis of generalized probabilistic theories, as they are shared by all of them. In this work, we explicitly characterise all possible mulitpartite measurements in the generalised probabilistic theory box-world for various numbers of parties n with systems characterised by n_i fiducial measurements (which can be thought of as inputs here) and n_o outcomes, for small n, n_i, n_o. This includes all n-party n_i-input, n_o-outcome wirings. For n > 2, we further classify these measurements into three classes: wirings, deterministic non-wiring type and non-deterministic non-wiring type measurements. We explore advantages of these different types of measurements over previous protocols in the context of non-locality distillation and state-distingishability. We further find examples of non-locality without entanglement (contrary to previous claims) and a relation of these measurements to classical process matrices.

"Imsets, introduced by Studený (see Studený, 2005 for details), are an algebraic method for representing conditional independence models. They have many attractive properties when applied to such models, and they are particularly nice when applied to directed acyclic graph (DAG) models. In particular, the standard imset for a DAG is in one-to-one correspondence with the independence model it induces, and hence is a label for its Markov equivalence class. We present a proposed extension to standard imsets for maximal ancestral graph (MAG) models, which have directed and bidirected edges, using the parameterizing set representation of Hu and Evans (2020). By construction, our imset also represents the Markov equivalence class of the MAG. We show that for many such graphs our proposed imset defines the model, though there is a subclass of graphs for which the representation does not. We prove that it does work for MAGs that include models with no adjacent bidirected edges, as well as for a large class of purely bidirected models. If there is time, we will also discuss applications of imsets to structure learning in MAGs. This is joint work with Zhongyi Hu (Oxford). References Z. Hu and R.J. Evans, Faster algorithms for Markov equivalence, In Proceedings for the 36th Conference on Uncertainty in Artificial Intelligence (UAI-2020), 2020. M. Studený, Probabilistic Conditional Independence Structures, Springer-Verlag, 2005."