Abstract
Estimation of sparse, high-dimensional precision matrices is an important and challenging problem. Existing methods all assume that observations can be made precisely but, in practice, this often is not the case; for example, the instruments used to measure the response may have limited precision. The present paper incorporates measurement error in the context of estimating a sparse, high-dimensional precision matrix. In particular, for a Gaussian graphical model with data corrupted by Gaussian measurement error with unknown variance, we establish a general result which gives sufficient conditions under which the posterior contraction rates that hold in the no-measurement-error case carry over to the measurement-error case. Interestingly, this result does not require that the measurement error variance be small. We apply our general result to several cases with well-known prior distributions for sparse precision matrices and also to a case with a newly-constructed prior for precision matrices with a sparse factor-loading form. Two different simulation studies highlight the empirical benefits of accounting for the measurement error as opposed to ignoring it, even when that measurement error is relatively small.
Highlights
The precision matrix, namely, the inverse of the covariance matrix of a Gaussian random vector, is a key object in multivariate analysis because of its role in describing conditional distributions
We focus here on a Gaussian measurement error model, for i = 1, . . . , n and j = 1, . . . , m, Yij = Xi + Zij, Xi ∼iid Np(0, Ω−1), Zij ∼iid Np(0, νIp) where the X and Z samples are mutually independent, Ip is the identity matrix of order p and m is the number of replicates for each X
We show below that failing to account for the measurement error creates a large bias and, certain adjustments are necessary to account for the presence of measurement error and to ensure accurate estimation of Ω
Summary
The precision matrix, namely, the inverse of the covariance matrix of a Gaussian random vector, is a key object in multivariate analysis because of its role in describing conditional distributions. Nghiem and McGee [5] assumed the variance of measurement error to be known, treated the unobservable outcomes as missing data and recently proposed a method to impute them and estimate the precision matrix iteratively They combined the imputation–regularized optimization algorithm [26] and Bayesian regularization for graphical models with unequal shrinkage [20] to formulate a new procedure and prove its consistency. We develop a general strategy that allows the user to incorporate additive Gaussian measurement error into existing Bayesian procedures for inference on structured, high-dimensional precision matrices in such a way that the posterior concentration rates are preserved and minimal changes to posterior computations are required.
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