Causal Discovery Under Latent Class Confounding: A Discrete Nonparametric Approach

Core Concepts

Discrete nonparametric causal discovery is possible under latent class confounding, by leveraging rank tests and coarsening of variables to overcome the limitations of conditional independence tests.

Abstract

The paper presents a novel approach for causal discovery in the presence of latent class confounding, where the relationships between observed variables are obscured by an unobserved global confounder.
Key highlights:

The authors show that despite the confounding effect of the latent variable, the causal structure can still be identifiable without making parametric assumptions, as long as the number of latent classes is small relative to the size and sparsity of the underlying directed acyclic graph (DAG).
They develop "rank tests" that can detect d-separation between variables by analyzing the rank of their joint probability matrices, without directly conditioning on the unobserved confounder.
To overcome the limitations of small-cardinality discrete variables, the authors introduce the concept of "coarsening" - creating supervariables of larger cardinality by grouping subsets of the original variables.
The algorithm proceeds in three phases: (1) using rank tests to remove edges, (2) leveraging the recovered sparsity to set up instances of discrete mixture model identification, and (3) propagating edge orientations.
Theoretical analysis shows the algorithm's runtime and sample complexity, as well as empirical results demonstrating its effectiveness on synthetic data.

Stats

The paper does not provide any specific numerical data or statistics. It focuses on the theoretical and algorithmic aspects of causal discovery under latent class confounding.

Quotes

"Discrete nonparametric causal discovery is possible under latent class confounding, so long as the number of latent classes remains small relative to the size and sparsity of the underlying DAG."
"We develop 'rank tests' that can detect d-separation between variables by analyzing the rank of their joint probability matrices, without directly conditioning on the unobserved confounder."
"To overcome the limitations of small-cardinality discrete variables, we introduce the concept of 'coarsening' - creating supervariables of larger cardinality by grouping subsets of the original variables."

Key Insights Distilled From

Discrete Nonparametric Causal Discovery Under Latent Class Confounding

by Bijan Mazahe... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2311.07454.pdf

Deeper Inquiries

How would the algorithm's performance scale with increasing dimensionality of the observed variables and the number of latent classes

The algorithm's performance would likely scale with increasing dimensionality of the observed variables and the number of latent classes. As the dimensionality of the observed variables increases, the number of possible coarsenings and separating sets also increases, leading to a larger search space for the algorithm. This could potentially increase the computational complexity and the number of rank tests required to detect non-adjacencies accurately. Similarly, as the number of latent classes increases, the algorithm would need to consider more possible sources of confounding, leading to a more complex mixture model to be identified. This could result in a higher number of k-MixProd runs and alignment steps, further increasing the computational burden of the algorithm.

What are the potential limitations or failure modes of the rank-based approach, and how could they be addressed

One potential limitation of the rank-based approach is the sensitivity to noise in the data, as singular matrices are unlikely to occur in empirical data. This could lead to challenges in accurately detecting non-adjacencies and could result in false positives or false negatives. To address this, a threshold value could be introduced to count the number of eigenvalues exceeding a certain threshold, providing a more robust measure of rank. Additionally, adjusting the threshold value could help in controlling the trade-off between sensitivity and specificity of the rank tests.
Another potential limitation could be the assumption of independence within source distributions, which may not always hold in real-world scenarios. This could lead to inaccuracies in the identification of separating sets and could impact the overall performance of the algorithm. One way to address this limitation could be to incorporate more sophisticated statistical techniques to model dependencies within source distributions, allowing for a more accurate representation of the underlying causal relationships.

Can the ideas presented in this work be extended to handle continuous variables or more complex causal structures beyond discrete DAGs

The ideas presented in this work could potentially be extended to handle continuous variables or more complex causal structures beyond discrete DAGs. For continuous variables, the rank-based approach could be adapted to work with continuous probability distributions, such as Gaussian distributions. This would involve modifying the rank tests to accommodate continuous data and developing algorithms that can handle the unique characteristics of continuous variables, such as correlation and variance.
To extend the approach to more complex causal structures beyond discrete DAGs, additional considerations would need to be made. This could involve incorporating more advanced modeling techniques, such as structural equation modeling, to capture the relationships between variables in a more nuanced way. Additionally, the algorithm could be enhanced to handle feedback loops, latent variables with continuous distributions, and non-linear relationships, allowing for a more comprehensive causal discovery process in complex systems.