MissNODAG: A Differentiable Framework for Learning Cyclic Causal Graphs from Incomplete Data, Including MNAR Mechanisms

Core Concepts

MissNODAG is a novel framework that effectively learns cyclic causal relationships from incomplete data, addressing limitations of existing methods by handling both MNAR missingness and feedback loops in systems.

Abstract

Bibliographic Information:

Sethuraman, M. G., Nabi, R., & Fekri, F. (2024). MissNODAG: Differentiable Cyclic Causal Graph Learning from Incomplete Data. arXiv preprint arXiv:2410.18918.

Research Objective:

This paper introduces MissNODAG, a novel framework designed to learn cyclic causal graphs from incomplete data, addressing the limitations of existing methods that struggle with feedback loops and MNAR (Missing Not At Random) data.

Methodology:

MissNODAG leverages an Expectation-Maximization (EM) algorithm to handle missing data. It alternates between imputing missing values and optimizing model parameters, incorporating:

• An additive noise model for causal relationships.
• Contractive residual flows to efficiently compute the log-determinant of the Jacobian matrix, ensuring tractability.
• Rejection sampling for imputation, with the option for direct sampling from the posterior distribution in specific cases (linear SEMs with MAR missingness).
• A block-parallel MNAR model for the missingness mechanism, allowing flexibility in handling real-world scenarios.

Key Findings:

• Through synthetic experiments, MissNODAG consistently outperforms state-of-the-art imputation techniques combined with causal learning on partially missing interventional data, demonstrating its superior performance in recovering both linear and nonlinear cyclic causal graphs.
• The framework effectively learns the underlying missingness mechanism, achieving high accuracy in recovering the m-graph edges, particularly with lower missingness probabilities.
• MissNODAG's performance is further validated through its application to a real-world gene regulatory network dataset, showcasing its practical relevance.

Main Conclusions:

MissNODAG presents a significant advancement in causal discovery by effectively handling both cyclic causal graphs and MNAR missingness, overcoming limitations of existing methods. Its ability to learn from incomplete data while accommodating feedback loops makes it a valuable tool for uncovering causal relationships in complex real-world systems.

Significance:

This research significantly contributes to the field of causal discovery by providing a robust and flexible framework for learning causal structures from incomplete data, which is a common challenge in many domains. MissNODAG's ability to handle both cyclic relationships and MNAR mechanisms broadens the applicability of causal discovery methods to more realistic scenarios.

Limitations and Future Research:

Future research directions include incorporating realistic measurement noise models, scaling the framework to larger graphs, allowing for unobserved confounders, and generalizing to broader classes of identifiable MNAR models.

Stats

The experiments used cyclic directed graphs with 10 nodes, generated using the Erdős-Rényi (ER) model with varying edge densities.
Sample sizes of 500 and 10000 were used for different experiments.
Missingness probabilities ranged from 0.1 to 0.5.
The performance was evaluated using Structural Hamming Distance (SHD).

Quotes

Key Insights Distilled From

MissNODAG: Differentiable Cyclic Causal Graph Learning from Incomplete Data

by Muralikrishn... at arxiv.org 10-25-2024

https://arxiv.org/pdf/2410.18918.pdf

Deeper Inquiries

How could MissNODAG be adapted to handle time-series data, where temporal dependencies are crucial for understanding causal relationships?

Adapting MissNODAG to handle time-series data, where temporal dependencies are paramount, would require several key modifications to effectively capture the dynamic causal relationships inherent in such data:

Dynamic Bayesian Network (DBN) Integration: Instead of static cyclic directed graphs, MissNODAG could be extended to leverage DBNs. DBNs are a natural fit for time-series data as they explicitly model the evolution of variables over time. This would involve representing the target law p(X) as a DBN, where nodes represent variables at different time points and edges encode temporal causal dependencies.

Temporal Missingness Mechanisms: The current MNAR model in MissNODAG assumes no temporal dependencies in the missingness mechanism. For time-series data, this assumption might be unrealistic. An extension could involve incorporating temporal dependencies in the missingness mechanism, allowing the probability of missingness at a particular time point to depend on both current and past values of variables, including past missingness indicators.

Recurrent Neural Networks for SEMs: To capture complex temporal dependencies in the structural equation models (SEMs), recurrent neural networks (RNNs), such as Long Short-Term Memory (LSTM) or Gated Recurrent Units (GRUs), could be employed. RNNs excel at modeling sequential data and can effectively learn the temporal relationships between variables in the time series.

Modified Optimization for Time Series: The optimization procedure would need adjustments to account for the temporal structure. Techniques like backpropagation through time (BPTT) could be used to train the RNN-based SEMs, while the EM algorithm could be adapted to handle the temporal dependencies in both the target law and the missingness mechanism.

By incorporating these modifications, MissNODAG could be effectively extended to handle time-series data, enabling the discovery of dynamic causal relationships even in the presence of missing data.

While MissNODAG demonstrates strong performance in handling MNAR data, could its reliance on specific MNAR models limit its generalizability to datasets with more complex or unknown missingness mechanisms?

You are right to point out that MissNODAG's reliance on specific MNAR models, particularly the block-parallel MNAR model, could potentially limit its generalizability to datasets with more complex or unknown missingness mechanisms.
Here's why:

Identifiability Assumptions: The block-parallel MNAR model makes specific assumptions about the relationships between missingness indicators and substantive variables, which are crucial for ensuring the identifiability of the full law. When these assumptions are violated, the EM algorithm might not converge to the true underlying model, leading to biased estimates.

Limited Scope of Mechanisms: Real-world datasets often exhibit missingness patterns that are more intricate than those captured by the block-parallel MNAR model. There might be complex interdependencies between missingness indicators or scenarios where missingness depends on unobserved confounders, which are not accounted for in the current framework.
To address these limitations and enhance generalizability, several research avenues could be explored:

Incorporating More General MNAR Models: Extending MissNODAG to handle a broader class of identifiable MNAR models, such as those with no colluders but allowing connections between missingness indicators, would be beneficial. This would require incorporating additional constraints during optimization to ensure the missingness mechanism forms a DAG and prevent convergence to non-identifiable solutions.

Non-parametric Missingness Mechanisms: Exploring non-parametric approaches to model the missingness mechanism could provide more flexibility in capturing complex dependencies. Techniques like Gaussian processes or deep generative models could be investigated to learn a more expressive representation of the missingness mechanism without relying on restrictive parametric assumptions.

Sensitivity Analysis: Incorporating sensitivity analysis techniques would be crucial to assess the robustness of MissNODAG's findings to deviations from the assumed MNAR model. This would involve systematically varying the missingness mechanism and evaluating the impact on the estimated causal graph, providing insights into the reliability of the results under different missingness scenarios.
By addressing these points, MissNODAG's applicability to a wider range of datasets with more complex and potentially unknown missingness mechanisms could be significantly improved.

Can the principles of causal discovery employed in MissNODAG be applied to other domains beyond traditional causal inference, such as reinforcement learning or natural language processing, to uncover hidden causal structures and improve decision-making in those fields?

Yes, the principles of causal discovery employed in MissNODAG, particularly its ability to handle cyclic dependencies and missing data, hold significant promise for application in domains beyond traditional causal inference, such as reinforcement learning (RL) and natural language processing (NLP), to uncover hidden causal structures and enhance decision-making.
Here's how these principles could be applied:
Reinforcement Learning:

Off-Policy Evaluation: In RL, evaluating the performance of a new policy without deploying it in the real world is crucial. MissNODAG's ability to handle missing data could be leveraged to estimate the reward of a target policy using data collected under a different behavior policy, even when the data collection mechanism is non-random (MNAR).

Causal World Models: Building causal world models that capture the causal relationships between actions, states, and rewards is essential for efficient RL. MissNODAG's ability to learn cyclic causal graphs could be valuable in identifying feedback loops and complex dependencies in the environment, leading to more accurate and robust world models.
Natural Language Processing:

Causal Text Analysis: Uncovering causal relationships between events and entities mentioned in text is crucial for tasks like event prediction, summarization, and question answering. MissNODAG's principles could be adapted to learn causal graphs from textual data, even when certain events or relationships are not explicitly stated (missing data).

Causal Reasoning in Dialogue Systems: Building dialogue systems capable of causal reasoning is essential for engaging in more meaningful and context-aware conversations. MissNODAG's ability to handle cyclic dependencies could be beneficial in modeling the complex interplay of utterances and their causal effects on dialogue flow and understanding.
Challenges and Considerations:

Domain-Specific Adaptations: Adapting MissNODAG to these domains would require careful consideration of domain-specific challenges. For instance, in RL, the sequential nature of data and the presence of feedback loops would necessitate modifications to the model architecture and optimization procedure.

Interpretability and Actionability: Ensuring the interpretability of the learned causal structures is crucial for decision-making in these domains. Techniques for visualizing and summarizing the causal relationships, as well as methods for quantifying uncertainty in the estimated causal effects, would be essential.
By addressing these challenges and leveraging the strengths of MissNODAG, we can unlock new possibilities for causal discovery in RL, NLP, and other domains, paving the way for more informed and effective decision-making in complex systems.