Dynamical systems are widely used in science and engineering to model systems consisting of several interacting components. Often, they can be given a causal interpretation in the sense that they not only model the evolution of the states of the system’s components over time, but also describe how their evolution is affected by external interventions on the system that perturb the dynamics. We introduce the formal framework of structural dynamical causal models (SDCMs) that explicates the causal semantics of the system’s components as part of the model. SDCMs represent a dynamical system as a collection of stochastic processes and specify the basic causal mechanisms that govern the dynamics of each...

Structural causal models (SCMs), also known as (nonparametric) structural equation models (SEMs), are widely used for causal modeling purposes. In particular, acyclic SCMs, also known as recursive SEMs, form a well-studied subclass of SCMs that generalize causal Bayesian networks to allow for latent confounders. In this paper, we investigate SCMs in a more general setting, allowing for the presence of both latent confounders and cycles. We show that in the presence of cycles, many of the convenient properties of acyclic SCMs do not hold in general: they do not always have a solution; they do not always induce unique observational, interventional and counterfactual distributions; a marginalization does not always exist,...

Structural Causal Models (SCMs) provide a popular causal modeling framework. In this work, we show that SCMs are not flexible enough to give a complete causal representation of dynamicat systems at equilibrium. Instead, we propose a generalization of the notion of an SCM, that we call Causal Constraints Model (CCM), and prove that CCMs do capture the causal semantics of such systems. We show how CCMs can be constructed from differential equations and initial conditions and we illustrate our ideas further on a simple but ubiquitous (bio)chemical reaction. Our framework also allows to model functional laws, such as the ideal gas law, in a sensible and intuitive way.

An important goal common to domain adaptation and causal inference is to make accurate predictions when the distributions for the source (or training) domain(s) and target (or test) domain(s) differ. In many cases, these different distributions can be modeled as different contexts of a single underlying system, in which each distribution corresponds to a different perturbation of the system, or in causal terms, an intervention. We focus on a class of such causal domain adaptation problems, where data for one or more source domains are given, and the task is to predict the distribution of a certain target variable from measurements of other variables in one or more target domains....

Structural Causal Models are widely used in causal modelling, but how they relate to other modelling tools is poorly understood. In this paper we provide a novel perspective on the relationship between Ordinary Differential Equations and Structural Causal Models. We show how, under certain conditions, the asymptotic behaviour of an Ordinary Differential Equation under non-constant interventions can be modelled using Dynamic Structural Causal Models. In contrast to earlier work, we study not only the effect of interventions on equilibrium states; rather, we model asymptotic behaviour that is dynamic under interventions that vary in time, and include as a special case the study of static equilibria.

Complex systems can be modelled at various levels of detail. Ideally, causal models of the same system should be consistent with one another in the sense that they agree in their predictions of the effects of interventions. We formalise this notion of consistency in the case of Structural Equation Models (SEMs) by introducing exact transformations between SEMs. This provides a general language to consider, for instance, the different levels of description in the following three scenarios: (a) models with large numbers of variables versus models in which the `irrelevant’ or unobservable variables have been marginalised out; (b) micro-level models versus macro-level models in which the macro-variables are aggregate features of...

The first part of this thesis provides an introduction to recent development in geometric quantization of symplectic and Poisson manifolds, including modern refinements involving Lie groupoid theory and index theory/K-theory. We start by giving a detailed treatment of traditional geometric quantization of symplectic manifolds, where we cover both the quantization scheme via polarization and via push-forward in K-theory. A different approach is needed for more general Poisson manifolds, which we treat by the geometric quantization of Poisson manifolds via the geometric quantization of their associated symplectic groupoids, due to Weinstein, Xu, Hawkins, et al. In the second part of the thesis we show that this geometric quantization via symplectic groupoids...