Foundations of Causal Discovery on Groups of Variables

TL;DR

This work focuses on widely employed assumptions for causal discovery when objects of interest are (multivariate) groups of random variables rather than individual (univariate) random variables, as is the case in a variety of problems in scientific domains such as climate science or neuroscience.

Abstract

Discovering causal relationships from observational data is a challenging task that relies on assumptions connecting statistical quantities to graphical or algebraic causal models. In this work, we focus on widely employed assumptions for causal discovery when objects of interest are (multivariate) groups of random variables rather than individual (univariate) random variables, as is the case in a variety of problems in scientific domains such as climate science or neuroscience. If the group-level causal models are derived from partitioning a micro-level model into groups, we explore the relationship between micro and group-level causal discovery assumptions. We investigate the conditions under which assumptions like Causal Faithfulness hold or fail to hold. Our analysis encompasses graphical causal models that contain cycles and bidirected edges. We also discuss grouped time series causal graphs and variants thereof as special cases of our general theoretical framework. Thereby, we aim to provide researchers with a solid theoretical foundation for the development and application of causal discovery methods for variable groups.

Figures (15)

• Figure 1: A mixed graph over micro-variables is coarsened to a mixed graph of variable groups, see Definition \ref{['def.coarsegraph']}.In graphical causal modelling, directed edges represent direct causal influences, bidirected edges represent confounding by a hidden variable and undirected edges indicate the presence of a selection variable that has been conditioned on.
• Figure 2: A diagrammatic summary of the relationship between the $\sigma$-Markov property and different types of faithfulness on a micro graph ${\mathcal{G}}$ and its graph of groups with respect to a partition ${\mathcal{P}}$, see Definition \ref{['def.graphs']} below for details. A blue arrow indicates that all properties from which the arrow emerges imply the target property. An orange arrow indicates that the properties from which the arrow emerges are not sufficient to guarantee the target property.
• Figure 3: Acyclification of a cyclic mixed graph to an acyclic mixed graph.
• Figure 4: Left: A DAG partitioned such that the resulting group DMG is cyclic, see also anand_causal_2023. Right: A cyclic micro DMG partitioned such that the resulting group DMG is acyclic.
• Figure 5: The micro path $\pi$ (in red) from $W_2$ to $Z_1$ is coarsened to the path $\mathrm{co}(\pi)$ in the group DMG $\mathrm{co}({\mathcal{G}},{\mathcal{P}}), \ {\mathcal{P}} = \{ {\mathbf{W}},{\mathbf{Y}},{\mathbf{Z}} \}$. The three ${\mathcal{P}}$-segments of $\pi$ are $W_2 \leftrightarrow W_1$, $Y_1 \leftarrow Y_3$ and $Z_1$.

Theorems & Definitions (62)

• Definition 1: m-separation, see zhang_causal_2008
• Definition 2: $\sigma$-separation, see forre_markov_2017
• Definition 3: Acyclification of a MG, see BonFOrPetMoo21
• Proposition 1
• Definition 4
• Definition 5: see anand_causal_2023
• Definition 6
• Definition 7
• Remark 1
• Lemma 1