I-CAM-UV: Integrating Causal Graphs over Non-Identical Variable Sets Using Causal Additive Models with Unobserved Variables

Hirofumi Suzuki; Kentaro Kanamori; Takuya Takagi; Thong Pham; Takashi Nicholas Maeda; Shohei Shimizu

cs.LGMar 3, 2026

I-CAM-UV: Integrating Causal Graphs over Non-Identical Variable Sets Using Causal Additive Models with Unobserved Variables

Authors: Hirofumi Suzuki, Kentaro Kanamori, Takuya Takagi, Thong Pham, Takashi Nicholas Maeda, Shohei Shimizu

Organizations: Fujitsu Limited · Faculty of Data Science, Shiga University · RIKEN AIP · Computer Centre, Gakushuin University ·

Abstract

Causal discovery from observational data is a fundamental tool in various fields of science. While existing approaches are typically designed for a single dataset, we often need to handle multiple datasets with non-identical variable sets in practice. One straightforward approach is to estimate a causal graph from each dataset and construct a single causal graph by overlapping. However, this approach identifies limited causal relationships because unobserved variables in each dataset can be confounders, and some variable pairs may be unobserved in any dataset. To address this issue, we leverage Causal Additive Models with Unobserved Variables (CAM-UV) that provide causal graphs having information related to unobserved variables. We show that the ground truth causal graph has structural consistency with the information of CAM-UV on each dataset. As a result, we propose an approach named I-CAM-UV to integrate CAM-UV results by enumerating all consistent causal graphs. We also provide an efficient combinatorial search algorithm and demonstrate the usefulness of I-CAM-UV against existing methods.

Figures & tables

Figure 1: Illustration of unobserved causal path (UCP) and unobserved backdoor path (UBP).

Figure 2: Example of CAM-UV. A UCP $v_{4}\rightarrow v_{5}\rightarrow v_{6}$ and a UBP $v_{2}\leftarrow v_{1}\rightarrow v_{3}$ are found. The dashed lines indicate undirected edges of the resulting mixed graph.

Figure 3: Example of inputs on a scenario we consider (left) and their simple overlapping (right).

Figure 4: Example of I-CAM-UV on the input of Figure 3 .

Figure 5: Example of I-CAM-UV resulting a unique DAG.

Figure 6: Examples of I-CAM-UV with relaxation.

Panel of Figure 7 — Figure 7: Accuracies on synthetic datasets.

Figure 8: Number of DAGs enumerated by I-CAM-UV.

Figure 9: Accuracy distributions on specific instances of synthetic datasets. The heatmaps show the distributions of DAGs output by I-CAM-UV.

Figure 10: Computation times on synthetic datasets.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 12 — Figure 12: Accuracies on fMRI dataset.

Figure 13: Number of DAGs enumerated by I-CAM-UV on fMRI dataset.

Panel of Figure 14 — Figure 14: Accuracy distributions on specific instances of fMRI dataset. The heatmaps show the distributions of DAGs output by I-CAM-UV. The top fours are cases of $b=0$ , the middle fours are cases of $b=1$ , and the bottom fours are cases of $b=2$ .

Figure 15: Computation times on fMRI dataset.

Figure 16: Minimum inconsistency costs on synthetic dataset.

Figure 17: Minimum inconsistency costs on fMRI dataset.

Figure 18: Status attainment model based on domain knowledge ( Duncan et al. 1972 ) . The solid directed edges represent expected causal relationships. The dashed bi-directed edges represent not modeled causal relationships, e.g., there may be latent confounders between the two, there may be a directed edge between the two, or the two may be independent.

Panel of Figure 19 — Figure 19: CAM-UV results for status attainment model dataset. The solid directed edges represent estimated causal relationships. The dashed edges represent not identified causal relationships due to the estimated existence of UCBs/UBPs.

Panel of Figure 20 — Figure 20: Three examples of the graphs enumerated by I-CAM-UV with specifying “Son’s Income” becomes the sink. The solid black directed edges represent correctly estimated causal relationships. The dashed red directed edges represent wrongly estimated causal relationships. The dotted blue directed edges represent expected but not estimated causal relationships.

Panel of Figure 21 — Figure 21: Three examples of the graphs enumerated by I-CAM-UV with specifying the following conditions: “Son’s Income” becomes the sink, and there is a causal relationship from “Father’s Education” to “Son’s Education”. The solid black directed edges represent correctly estimated causal relationships. The dashed red directed edges represent wrongly estimated causal relationships. The dotted blue directed edges represent expected but not estimated causal relationships.

Panel of Figure 22 — Figure 22: Three examples of the graphs enumerated by I-CAM-UV with specifying the following conditions: “Son’s Income” becomes the sink, there is a causal relationship from “Father’s Education” to “Son’s Education”, and there is a causal relationship from “Father’s Occupation” to “Son’s Education”. The solid black directed edges represent correctly estimated causal relationships. The dashed red directed edges represent wrongly estimated causal relationships. The dotted blue directed edges represent expected but not estimated causal relationships.

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