A Meta-Learning Approach to Bayesian Causal Discovery (2412.16577v3)

Abstract: Discovering a unique causal structure is difficult due to both inherent identifiability issues, and the consequences of finite data. As such, uncertainty over causal structures, such as those obtained from a Bayesian posterior, are often necessary for downstream tasks. Finding an accurate approximation to this posterior is challenging, due to the large number of possible causal graphs, as well as the difficulty in the subproblem of finding posteriors over the functional relationships of the causal edges. Recent works have used meta-learning to view the problem of estimating the maximum a-posteriori causal graph as supervised learning. Yet, these methods are limited when estimating the full posterior as they fail to encode key properties of the posterior, such as correlation between edges and permutation equivariance with respect to nodes. Further, these methods also cannot reliably sample from the posterior over causal structures. To address these limitations, we propose a Bayesian meta learning model that allows for sampling causal structures from the posterior and encodes these key properties. We compare our meta-Bayesian causal discovery against existing Bayesian causal discovery methods, demonstrating the advantages of directly learning a posterior over causal structure.

• The paper introduces Bayesian Causal Neural Process (BCNP), a novel meta-learning model for Bayesian causal discovery directly learning posterior distributions over causal structures.
• BCNP employs a permutation equivariant encoder-decoder architecture that allows direct sampling of directed acyclic graphs (DAGs) through a new matrix parameterization.
• Experiments demonstrate BCNP outperforms existing methods across various datasets and metrics, effectively handling challenges like limited data and unidentifiable causal directions.

A Meta-Learning Approach to Bayesian Causal Discovery

The paper, "A Meta-Learning Approach to Bayesian Causal Discovery," addresses the challenge of inferring causal structures from observational data, a task complicated by identifiability issues and limited data availability. The authors propose a novel Bayesian meta-learning model designed to overcome these challenges by directly learning the posterior distribution over causal structures. This approach improves upon previous methods by incorporating critical properties like permutation equivariance and dependencies between edges and enables direct sampling of directed acyclic graphs (DAGs).

Summary of the Approach

The proposed method reframes the problem of causal discovery as a supervised learning task using meta-learning. By doing so, it circumvents the traditional need to approximate posteriors over complex functional relationships between variables, a process that can be computationally intensive and difficult, particularly for non-linear relationships modeled by neural networks.

The core of the model, named Bayesian Causal Neural Process (BCNP), is an encoder-decoder architecture. The encoder processes datasets, ensuring that the constructed representations are permutation invariant regarding sample order and permutation equivariant concerning node order. This is crucial as it allows the model to generalize across different causal structures without being biased by node arrangements within the datasets.

The paper makes a significant contribution with its decoder, which differs from past approaches by directly sampling DAGs through a novel parameterization using permutation and lower triangular matrices. This parameterization allows the model to produce valid causal structures inherently, avoiding issues associated with cyclic graphs or disconnected structures often observed in prior work.

Key Results and Comparisons

The experimental results demonstrate that BCNP effectively approximates the true posterior in synthetic datasets where the causal relationships are known. This capability is crucial for its applicability in real-world scenarios where causal inference can guide critical decision-making processes.

Experiments conducted on various datasets highlight BCNP's superiority over traditional explicit Bayesian models like DiBS and BayesDAG and other meta-learning models such as AVICI and CSIvA. In particular, BCNP outperforms these methods significantly on metrics such as the AUC, log probability, and expected edge F1 scores while maintaining competitive expected Structural Hamming Distance (SHD) scores. This performance is consistent across various types of data generation processes, including linear, neural network-based, and Gaussian Process Conditional Density Estimation (GPCDE) functions.

Furthermore, the model's ability to handle unidentifiable cases is noteworthy. In evaluations with two-variable Gaussian data, where causal directions are not identifiable, BCNP robustly handles the uncertainty, highlighting its strength in providing sound causal structures.

Theoretical and Practical Implications

The introduction of BCNP has meaningful implications for both theoretical advancements and practical applications. Theoretically, it provides a robust framework for Bayesian causal discovery by directly learning from data distributions, thus potentially bridging gaps between observational datasets and causal inference challenges.

Practically, the capacity to sample directly from posterior distributions over causal structures opens doors to more sophisticated analysis in fields such as genomics, epidemiology, and economics, where underlying causal relationships operate over complex and interdependent networks.

Future Directions

The authors suggest that the BCNP model could be further enhanced by incorporating interventional data, thereby improving causal structure learning outcomes. Beyond this, the methodology's adaptability to different causal discovery tasks and datasets with varied underlying data generating processes suggests a wide field of future research and application. As AI systems continue to evolve, methodologies like BCNP will likely play a pivotal role in the advancement of causal inference technology, offering robust solutions and insights in data-driven domains.