[Paper Review] Model-Preserving Sensitivity Analysis for Families of Gaussian Distributions

Christiane Goergen, Manuele Leonelli|arXiv (Cornell University)|Sep 27, 2018

Bayesian Modeling and Causal Inference22 references4 citations

TL;DR

This paper introduces model-preserving sensitivity analysis for Gaussian graphical models, ensuring that perturbations to covariance parameters maintain the original conditional independence structure by using multiplicative, algebraically constrained covariation. The method preserves the graphical model's validity while quantifying distributional changes via Kullback-Leibler divergence, showing robustness comparable to standard methods but with coherent conditional independence relationships.

ABSTRACT

The accuracy of probability distributions inferred using machine-learning algorithms heavily depends on data availability and quality. In practical applications it is therefore fundamental to investigate the robustness of a statistical model to misspecification of some of its underlying probabilities. In the context of graphical models, investigations of robustness fall under the notion of sensitivity analyses. These analyses consist in varying some of the model's probabilities or parameters and then assessing how far apart the original and the varied distributions are. However, for Gaussian graphical models, such variations usually make the original graph an incoherent representation of the model's conditional independence structure. Here we develop an approach to sensitivity analysis which guarantees the original graph remains valid after any probability variation and we quantify the effect of such variations using different measures. To achieve this we take advantage of algebraic techniques to both concisely represent conditional independence and to provide a straightforward way of checking the validity of such relationships. Our methods are demonstrated to be robust and comparable to standard ones, which break the conditional independence structure of the model, using an artificial example and a medical real-world application.

Motivation & Objective

• To address the limitation in standard Gaussian sensitivity analysis where perturbations to the covariance matrix break the original conditional independence structure.
• To develop a sensitivity analysis framework that preserves the graphical model's structure after parameter variation.
• To ensure that all perturbed distributions remain valid Gaussian models with the same conditional independence relationships as the original.
• To provide a continuous analogue to proportional covariation in discrete Bayesian networks, maintaining model coherence.
• To demonstrate the robustness and practical utility of the method through artificial and real-world medical examples.

Proposed method

• Propose model-preserving perturbations that act multiplicatively on covariance matrix entries rather than additively.
• Use algebraic techniques to represent and verify conditional independence structures via the vanishing of specific minors in the covariance matrix.
• Enforce covariation of additional parameters to preserve the original graph structure after perturbation.
• Implement five covariation schemes: full, partial, row-based, column-based, and standard variation, each ensuring the resulting matrix remains a valid covariance matrix.
• Quantify the effect of perturbations using Kullback-Leibler divergence and Frobenius norm between original and varied distributions.
• Leverage Schur products and positive semidefiniteness checks to validate the resulting covariance matrices.

Experimental results

Research questions

• RQ1 Can we perform sensitivity analysis on Gaussian graphical models without disrupting the original conditional independence structure?
• RQ2 How do model-preserving perturbations compare to standard additive perturbations in terms of KL divergence and model coherence?
• RQ3 Which covariation scheme (e.g., row-based, column-based) leads to the most stable and representative sensitivity results?
• RQ4 Under what conditions is the resulting covariance matrix after model-preserving perturbation guaranteed to remain positive semidefinite?
• RQ5 How does the proposed method perform in real-world applications compared to standard sensitivity analysis?

Key findings

• Model-preserving sensitivity analysis maintains the original graphical structure after perturbation, ensuring the graph remains a faithful representation of the model’s conditional independence structure.
• In the Cachexia patient network, traditional methods produced significantly smaller KL divergences than model-preserving methods, but the latter showed more consistent results across different parameters.
• Row-based model-preserving variations led to significantly smaller KL divergences in two out of four cases, indicating greater sensitivity to certain covariances.
• For the Glutamine/Betaine covariance, the row-based model-preserving scheme achieved a Frobenius norm nearly equal to that of the standard method, demonstrating strong performance.
• The full model-preserving covariation scheme always results in a positive semidefinite matrix, unlike partial or row/column-based schemes which require explicit checks.
• The method enables robust sensitivity analysis with coherent graphical representations, offering a viable alternative to standard approaches that compromise model structure.