A Conditional Independence Test for Latent Gaussian Variables with Discretized Observations

Core Concepts

The core message of this paper is to propose a novel conditional independence test, DCT, that can accurately assess the conditional independence relationships among latent Gaussian variables when only their discretized observations are available.

Abstract

The paper addresses the problem of testing conditional independence in the presence of discretization. Existing conditional independence tests assume direct access to observations from all variables and can lead to incorrect conclusions when some variables are discretized. The key contributions are: Developing bridge equations to effectively estimate the underlying conditional independence from discretized observations. The bridge equations connect the discretized observations to the parameters of the latent Gaussian model. Deriving appropriate test statistics and their asymptotic distributions under the null hypotheses of unconditional independence and conditional independence. This allows for valid statistical inference. Demonstrating the versatility of the proposed DCT test, which can handle various scenarios - when both observed variables are continuous, both are discretized, or one is continuous and the other is discretized. Providing theoretical results and empirical validation to show the effectiveness of the DCT test in accurately assessing conditional independence in the presence of discretization, outperforming existing methods. The paper first introduces the problem setting and the preliminary framework of the DCT test. It then details the design of the bridge equations for different cases, followed by the derivation of the test statistics and their asymptotic distributions for both unconditional and conditional independence testing. Finally, experimental results on synthetic data are presented to evaluate the performance of the DCT test.

Stats

The proportion of observations where both ˜Xj1 and ˜Xj2 are greater than their respective means: ˆτj1,j2 = 1/n Σni=1 1{˜xij1 > Pn˜Xj1, ˜xij2 > Pn˜Xj2}. The proportion of observations where ˜Xj is greater than its mean: ˆτj = 1/n Σni=1 1{˜xij > Pn˜Xj}. The sample covariance between Xj1 and Xj2: ˆσj1,j2 = 1/n Σni=1 xij1xij2 - 1/n Σni=1 xij1 1/n Σni=1 xij2.

Quotes

"Directly applying existing conditional independence tests to those discretized observations is very likely to derive the wrong conclusion." "When X1 and X3 are transformed into their discretized forms, ˜X1 and ˜X3, this embedded information about X2 persists. Consequently, ˜X1 and ˜X3, containing shared information about X2, is dependent on each other when conditioned on ˜X2 (˜X1 and ˜X3 are d-connected given ˜X2)."

Key Insights Distilled From

A Conditional Independence Test in the Presence of Discretization

by Boyang Sun,Y... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.17644.pdf

Deeper Inquiries

How can the proposed DCT test be extended to handle non-Gaussian latent variables or nonlinear relationships between the variables?

The proposed DCT test can be extended to handle non-Gaussian latent variables or nonlinear relationships between the variables by incorporating kernel methods or non-parametric approaches. Kernel methods, such as kernel conditional independence tests, can capture nonlinear relationships between variables by mapping them into a higher-dimensional space where linear relationships may hold. By utilizing kernel methods, the DCT test can effectively handle non-Gaussian latent variables and nonlinear dependencies between variables. Additionally, incorporating non-parametric methods that do not assume a specific parametric form for the data distribution can also enhance the test's ability to capture complex relationships in the data.

What are the potential limitations of the binarization approach used in the DCT test, and how can it be improved to better utilize the information in the discretized observations?

One potential limitation of the binarization approach used in the DCT test is the loss of information due to the discretization process. Binarization reduces the continuous variables to binary values, leading to a loss of granularity and potentially important information contained in the original data. To improve the utilization of information in discretized observations, the DCT test can explore alternative discretization methods that preserve more information, such as using multiple levels of discretization instead of binary values. By incorporating more levels of discretization, the test can capture a wider range of relationships between variables and reduce information loss.

Can the DCT test be adapted to work in a distributed or federated learning setting, where the data is spread across multiple locations with different discretization schemes?

Yes, the DCT test can be adapted to work in a distributed or federated learning setting by incorporating techniques for handling data spread across multiple locations with different discretization schemes. One approach is to standardize the discretization process across all locations to ensure consistency in the data representation. Alternatively, the DCT test can be modified to accommodate different discretization schemes by incorporating a data harmonization step that aligns the discretization levels before conducting the conditional independence test. By addressing the challenges of varying discretization schemes in distributed settings, the DCT test can effectively analyze data from multiple locations while maintaining the integrity of the results.