Causality and independence in systems of equations

The technique of causal ordering is used to study causal and probabilistic aspects implied by model equations. Causal discovery algorithms are used to learn causal and dependence structure from data. In this thesis, 'Causality and independence in systems of equations', we explore the relationship between causal ordering and the output of causal discovery algorithms. By combining these techniques, we bridge the gap between the world of dynamical systems at equilibrium and literature regarding causal methods for static systems. In a nutshell, this gives new insights about models with feedback and an improved understanding of observed phenomena in certain (biological) systems. Based on our ideas, we outline a novel approach towards causal discovery for dynamical systems at equilibrium. This work was inspired by a desire to understand why the output of causal discovery algorithms sometimes appears to be at odds with expert knowledge. We were particularly interested in explaining apparent reversals of causal directions when causal discovery methods are applied to protein expression data. We propose the presence of a perfectly adapting feedback mechanism or unknown measurement error as possible explanations for these apparent reversals. We develop conditions for the detection of perfect adaptation from model equations or from data and background knowledge. This can be used to reason about the existence of feedback mechanisms using only partial observations of a system, resulting in additional criteria for data-driven selection of causal models. This line of research was made possible by novel interpretations and extensions of the causal ordering algorithm. Additionally, we challenge a key assumption in many causal discovery algorithms; that the underlying system can be modelled by the well-known class of structural causal models. To overcome the limitations of these models in capturing the causal semantics of dynamical systems at equilibrium, we propose a generalization that we call causal constraints models. Looking beyond standard causal modelling frameworks allows us to further explore the relationship between dynamical models at equilibrium and methods for causal discovery on equilibrium data