2021/07/23 by Noirrit Kiran Chandra, Peter Müeller, Chandra, Noirrit Kiran +4 · 1 citation
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Bayesian Modeling and Causal Inference #FOS: Computer and information sciences #Methodology (stat.ME) #Statistical Methods and Bayesian Inference #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.2107.11316
openalex publication_date 2021/07/23 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We propose a novel approach to estimating the precision matrix of\nmultivariate Gaussian data that relies on decomposing them into a low-rank and\na diagonal component. Such decompositions are very popular for modeling large\ncovariance matrices as they admit a latent factor based representation that\nallows easy inference. The same is however not true for precision matrices due\nto the lack of computationally convenient representations which restricts\ninference to low-to-moderate dimensional problems. We address this remarkable\ngap in the literature by building on a latent variable representation for such\ndecomposition for precision matrices. The construction leads to an efficient\nGibbs sampler that scales very well to high-dimensional problems far beyond the\nlimits of the current state-of-the-art. The ability to efficiently explore the\nfull posterior space also allows the model uncertainty to be easily assessed.\nThe decomposition crucially additionally allows us to adapt sparsity inducing\npriors to shrink the insignificant entries of the precision matrix toward zero,\nmaking the approach adaptable to high-dimensional small-sample-size sparse\nsettings. Exact zeros in the matrix encoding the underlying conditional\nindependence graph are then determined via a novel posterior false discovery\nrate control procedure. A near minimax optimal posterior concentration rate for\nestimating precision matrices is attained by our method under mild regularity\nassumptions. We evaluate the method's empirical performance through synthetic\nexperiments and illustrate its practical utility in data sets from two\ndifferent application domains.\n