2023/05/23 by P.S. Sarkar, Sarkar, Partha, Kshitij Khare +3 · 1 citation
Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Statistical Distribution Estimation and Applications #Statistical Methods and Bayesian Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2305.13743
openalex publication_date 2023/05/23 · openalex created_date 2023/05/27 · openalex updated_date 2026/07/28
The Inverse-Wishart (IW) distribution is a standard and popular choice of priors for covariance matrices and has attractive properties such as conditional conjugacy. However, the IW family of priors has crucial drawbacks, including the lack of effective choices for non-informative priors. Several classes of priors for covariance matrices that alleviate these drawbacks, while preserving computational tractability, have been proposed in the literature. These priors can be obtained through appropriate scale mixtures of IW priors. However, in the era of increasing dimensionality, the posterior consistency of models that incorporate such priors has not been investigated. We address this issue for the multi-response regression setting (q responses, n samples) under a wide variety of IW scale mixture priors for the error covariance matrix. Posterior consistency and contraction rates for both the regression coefficient matrix and the error covariance matrix are established in the ``large q, large n'' setting under mild assumptions on the true data-generating covariance matrix and relevant hyperparameters. In particular, the number of responses qn is allowed to grow with n, but with qn = o(n). Also, some results related to the inconsistency of the posterior distribution and posterior mean for qn/n → γ, where γ∈ (0,∞) are provided.