An identity for multidimensional continuous exponential families and its applications
Journal of Multivariate Analysis
Fundamentals of statistical signal processing: estimation theory
Fundamentals of statistical signal processing: estimation theory
Journal of Multivariate Analysis
Efficient Stepwise Selection in Decomposable Models
UAI '01 Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence
A well-conditioned estimator for large-dimensional covariance matrices
Journal of Multivariate Analysis
On improved estimation of normal precision matrix and discriminant coefficients
Journal of Multivariate Analysis
Walk-Sums and Belief Propagation in Gaussian Graphical Models
The Journal of Machine Learning Research
Factored sparse inverse covariance matrices
ICASSP '00 Proceedings of the Acoustics, Speech, and Signal Processing, 2000. on IEEE International Conference - Volume 02
The Journal of Machine Learning Research
Rethinking Biased Estimation: Improving Maximum Likelihood and the Cramér–Rao Bound
Foundations and Trends in Signal Processing
Multiscale Gaussian Graphical Models and Algorithms for Large-Scale Inference
SSP '07 Proceedings of the 2007 IEEE/SP 14th Workshop on Statistical Signal Processing
Generalized SURE for exponential families: applications to regularization
IEEE Transactions on Signal Processing
Detection of Gauss-Markov random fields with nearest-neighbor dependency
IEEE Transactions on Information Theory
Decomposable principal component analysis
IEEE Transactions on Signal Processing
Time-Varying Autoregressive (TVAR) Models for Multiple Radar Observations
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing
Embedded trees: estimation of Gaussian Processes on graphs with cycles
IEEE Transactions on Signal Processing
Dimension Estimation in Noisy PCA With SURE and Random Matrix Theory
IEEE Transactions on Signal Processing
Estimation in Gaussian Graphical Models Using Tractable Subgraphs: A Walk-Sum Analysis
IEEE Transactions on Signal Processing
Matrices with banded inverses: inversion algorithms and factorization of Gauss-Markov processes
IEEE Transactions on Information Theory
A New SURE Approach to Image Denoising: Interscale Orthonormal Wavelet Thresholding
IEEE Transactions on Image Processing
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Graphical models are a framework for representing and exploiting prior conditional independence structures within distributions using graphs. In the Gaussian case, these models are directly related to the sparsity of the inverse covariance (concentration) matrix and allow for improved covariance estimation with lower computational complexity. We consider concentration estimation with the mean-squared error (MSE) as the objective, in a special type of model known as decomposable. This model includes, for example, the well known banded structure and other cases encountered in practice. Our first contribution is the derivation and analysis of the minimum variance unbiased estimator (MVUE) in decomposable graphical models. We provide a simple closed form solution to the MVUE and compare it with the classical maximum likelihood estimator (MLE) in terms of performance and complexity. Next, we extend the celebrated Stein's unbiased risk estimate (SURE) to graphical models. Using SURE, we prove that the MSE of the MVUE is always smaller or equal to that of the biased MLE, and that the MVUE itself is dominated by other approaches. In addition, we propose the use of SURE as a constructive mechanism for deriving new covariance estimators. Similarly to the classical MLE, all of our proposed estimators have simple closed form solutions but result in a significant reduction in MSE.