Clustering and Pruning in Causal Data Fusion

TL;DR

The paper introduces pruning and clustering as principled preprocessing steps for causal data fusion with multiple data sources, addressing the scalability challenges of do-calculus-based identification. It establishes formal conditions under which these size-reduction operations preserve identifiability and provide ways to transfer identifying functionals back to the original graphs. Through theoretical results, algorithmic procedures (VerifyInputs), and demonstrations on epidemiological and social science data, the work shows that clustering and pruning can dramatically reduce computation time while maintaining correct identification conclusions. Simulations indicate substantial runtime gains in moderately sized graphs, highlighting practical benefits for complex causal data-fusion problems. The framework also includes counterexamples illustrating the necessity of certain conditions for identification invariance and offers guidance for applying these techniques in planning data collection.

Abstract

Data fusion, the process of combining observational and experimental data, can enable the identification of causal effects that would otherwise remain non-identifiable. Although identification algorithms have been developed for specific scenarios, do-calculus remains the only general-purpose tool for causal data fusion, particularly when variables are present in some data sources but not others. However, approaches based on do-calculus may encounter computational challenges as the number of variables increases and the causal graph grows in complexity. Consequently, there exists a need to reduce the size of such models while preserving the essential features. For this purpose, we propose pruning (removing unnecessary variables) and clustering (combining variables) as preprocessing operations for causal data fusion. We generalize earlier results on a single data source and derive conditions for applying pruning and clustering in the case of multiple data sources. We give sufficient conditions for inferring the identifiability or non-identifiability of a causal effect in a larger graph based on a smaller graph and show how to obtain the corresponding identifying functional for identifiable causal effects. Examples from epidemiology and social science demonstrate the use of the results.

Figures (11)

• Figure 1: Illustrations of size-reduction techniques for causal graphs: \ref{['fig:size1']} the original graph, \ref{['fig:size3']} a pruned graph where $Z_5$ and $Z_6$ have been deemed irrelevant for the identification of $p(y \,\vert\, \mathrm{do}(x))$ and thus removed, \ref{['fig:size4']} a clustered graph obtained from the pruned graph where $T$ represents the set $\{Z_1,Z_2,Z_3\}$.
• Figure 2: Causal graphs demonstrating the necessity of the assumption $\mathbf{B}_i \cup \mathbf{C}_i \subset \mathrm{an} _\mathcal{G}(\mathbf{Y})$ for pruning non-ancestors.
• Figure 3: Graphs related to the example on pruning with multiple data sources: \ref{['fig:graph2g']} the original graph, \ref{['fig:graph2gprime']} the pruned graph obtained by applying Theorems \ref{['thm:Yancestors']}, \ref{['thm:Xancestors']}, and \ref{['thm:isolated']}.
• Figure 4: Graphs for the example on clustering with multiple data sources: \ref{['fig:clustg']} the original graph, \ref{['fig:clustgdot']} the clustered graph where the sets $\{R, S_1, S_2, E_1, E_2\}$ and $\{T_1, T_2\}$ have been clustered as $S$ and $T$, respectively.
• Figure 5: Causal graphs demonstrating the importance of compatibility of input distributions for the identification invariance of clustering.

Theorems & Definitions (11)

• Definition 1: Structural causal model
• Definition 2: d-separation
• Definition 3: Submodel
• Definition 4: Identifiability
• Definition 5: Identification invariance
• Definition 6: Restricted input distributions
• Definition 7: Clustering
• Definition 8: Receiver
• Definition 9: Emitter
• Definition 10: Transit cluster