From Deterministic ODEs to Dynamic Structural Causal Models

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.

[1]  Denver Dash,et al.  Restructuring Dynamic Causal Systems in Equilibrium , 2005, AISTATS.

[2]  S. Bittanti,et al.  Periodic Systems: Filtering and Control , 2008 .

[3]  Herbert A. Simon,et al.  Causality and Model Abstraction , 1994, Artif. Intell..

[4]  Joris M. Mooij,et al.  Structural Causal Models: Cycles, Marginalizations, Exogenous Reparametrizations and Reductions , 2016, ArXiv.

[5]  Alexander Sokol,et al.  Causal interpretation of stochastic differential equations , 2013, 1304.0217.

[6]  David Card,et al.  Minimum Wages and Employment: A Case Study of the Fast Food Industry in New Jersey and Pennsylvania , 1993 .

[7]  Peter Spirtes,et al.  Directed Cyclic Graphical Representations of Feedback Models , 1995, UAI.

[8]  Frederick Eberhardt,et al.  Learning linear cyclic causal models with latent variables , 2012, J. Mach. Learn. Res..

[9]  Marek J. Druzdzel,et al.  Learning Why Things Change: The Difference-Based Causality Learner , 2010, UAI.

[10]  Bernhard Schölkopf,et al.  On Causal Discovery with Cyclic Additive Noise Models , 2011, NIPS.

[11]  Herbert A. Simon,et al.  Causality in Bayesian Belief Networks , 1993, UAI.

[12]  Patrik O. Hoyer,et al.  Discovering Cyclic Causal Models by Independent Components Analysis , 2008, UAI.

[13]  Andreas Ritter,et al.  Structural Equations With Latent Variables , 2016 .

[14]  Bernhard Schölkopf,et al.  From Ordinary Differential Equations to Structural Causal Models: the deterministic case , 2013, UAI.

[15]  Nir Friedman,et al.  Probabilistic Graphical Models: Principles and Techniques - Adaptive Computation and Machine Learning , 2009 .

[16]  J. Pearl.  Causality: Models, Reasoning and Inference , 2000 .

[17]  C. Granger.  Investigating causal relations by econometric models and cross-spectral methods , 1969 .

[18]  Bernhard Schölkopf,et al.  Theoretical Aspects of Cyclic Structural Causal Models , 2016 .

[19]  Joris M. Mooij,et al.  From Random Differential Equations to Structural Causal Models: the stochastic case , 2018, ArXiv.