Bayesian Regularization of Gaussian Graphical Models with Measurement Error

Abstract

We consider a framework for determining and estimating the conditional pairwise relationships of variables when the observed samples are contaminated with measurement error in high dimensional settings. Assuming the true underlying variables follow a multivariate Gaussian distribution, if no measurement error is present, this problem is often solved by estimating the precision matrix under sparsity constraints. However, when measurement error is present, not correcting for it leads to inconsistent estimates of the precision matrix and poor identification of relationships. We propose a new Bayesian methodology to correct for the measurement error from the observed samples.This Bayesian procedure utilizes a recent variant of the spike-and-slab Lasso to obtain a point estimate of the precision matrix, and corrects for the contamination via the recently proposed Imputation-Regularization Optimization procedure designed for missing data. Our method is shown to perform better than the naive method that ignores measurement error in both identification and estimation accuracy. To show the utility of the method, we apply the new method to establish a conditional gene network from a microarray dataset.

A Simple Correction Procedure for High-Dimensional Generalized Linear Models with Measurement Error

“…With mild conditions on the regularization procedure and variability of the data, the findings of [Liang et al, 2018] show that the IRO-algorithm gives a consistent estimate of the optimized parameters in each iteration. Moreover, the results of [Byrd et al, 2019] extend this result to the measurement error scenario.…”

“…We make use of the already processed output for the same dataset found in [Nghiem and Potgieter, 2018] and [Byrd et al, 2019]. After removing genes with es-timated signal-to-noise ratio larger than 0.5, there were a remaining p = 2074 genes remaining.…”

“…. , n. Recently, the IRO-algorithm was used in a measurement error correction procedure for Gaussian graphical models, which estimates the precision matrix Ω x with an assumed known or estimable Ω u [Byrd et al, 2019]. Going forward we will refer to a procedure using the IRO-algorithm to correct for measurement error as an IRO-adjusted procedure.…”

“…The IRO-algorithm was proposed as a technique for missing data in the high-dimensional setting, which notably gives a flexible framework with consistency guarantees for latent variables. Recently, the IRO-algorithm was used in the context of estimating Gaussian graphical models with mismeasured observations [Byrd et al, 2019]. The procedure was shown to be asymptomatically consistent; in addition it greatly reduced the number of false positives found in the selection process and reduced the overall estimation error.…”