Title: Marginalization and reduction of structural causal models
Authors: Stephan Bongers - Delft University of Technology (Netherlands) [presenting]
Bernhard Scholkopf - Max Planck Institute for Intelligent Systems (Germany)
Joris Mooij - University of Amsterdam (Netherlands)
Abstract: Structural Causal Models, also known as (Non-Parametric) Structural Equation Models, are widely used for causal modelling purposes. One of their advantages is that they allow for cycles, i.e., causal feedback loops. In this work, we give a rigorous treatment of Structural Causal Models. Two different types of variables play a role in SCMs: ``endogenous'' variables and ``exogenous'' variables (also known as ``disturbance terms''). We define a marginalization operation on SCMs that effectively removes a subset of the endogenous variables from the model. This operation can be seen as projecting the description of a full system to the description of a subsystem. We show that this operation preserves the causal semantics. We also show that in the linear case, the number of exogenous variables can be reduced so that only a single one-dimensional disturbance term is needed for each endogenous variable. This ``reduction'' can reduce the model complexity significantly and offers parsimonious representations for the linear case. In general, we show that it is always possible to reduce to only one single one-dimensional disturbance term, however this comes at the price of loosing dimensional information. We show that under certain additional conditions such a reduction is not possible.