Sensitivity Analysis for Inference with Partially Identifiable Covariance Matrices

TL;DR

This paper introduces a semidefinite programming method to quantify bias in covariance matrix inference when data is partially missing, providing more accurate error assessments than traditional sampling uncertainty estimates.

Contribution

It develops a novel approach to automatically quantify bias in partially identifiable covariance matrices using convex optimization techniques.

Findings

Method accurately assesses true error in missing data imputation.

Provides bounds on inference bias due to partial identifiability.

Outperforms sampling-based uncertainty estimates in bias quantification.

Abstract

In some multivariate problems with missing data, pairs of variables exist that are never observed together. For example, some modern biological tools can produce data of this form. As a result of this structure, the covariance matrix is only partially identifiable, and point estimation requires that identifying assumptions be made. These assumptions can introduce an unknown and potentially large bias into the inference. This paper presents a method based on semidefinite programming for automatically quantifying this potential bias by computing the range of possible equal-likelihood inferred values for convex functions of the covariance matrix. We focus on the bias of missing value imputation via conditional expectation and show that our method can give an accurate assessment of the true error in cases where estimates based on sampling uncertainty alone are overly optimistic.