# Bayesian Sparse Covariance Estimation Provides a Bayesian sparse and positive definite estimate of a covariance matrix via the beta-mixture shrinkage prior. ```r bmspcov(X, Sigma, prior = list(), nsample = list()) ``` ## Arguments - `X`: a n $\times$ p data matrix with column mean zero. - `Sigma`: an initial guess for Sigma. - `prior`: a list giving the prior information. The list includes the following parameters (with default values in parentheses): `a (1/2)` and `b (1/2)` giving the shape parameters for beta distribution, `lambda (1)` giving the hyperparameter for the diagonal elements, `tau1sq (10000/(n*p^4))` giving the hyperparameter for the shrinkage prior of covariance. - `nsample`: a list giving the MCMC parameters. The list includes the following integers (with default values in parentheses): `burnin (1000)` giving the number of MCMC samples in transition period, `nmc (1000)` giving the number of MCMC samples for analysis. ## Returns - **Sigma**: a nmc $\times$ p(p+1)/2 matrix including lower triangular elements of covariance matrix. - **Phi**: a nmc $\times$ p(p+1)/2 matrix including shrinkage parameters corresponding to lower triangular elements of covariance matrix. - **p**: dimension of covariance matrix. ## Details Lee, Jo and Lee (2022) proposed the beta-mixture shrinkage prior for estimating a sparse and positive definite covariance matrix. The beta-mixture shrinkage prior for $\Sigma = (\sigma_{jk})$ is defined as $$ \pi(\Sigma) = \frac{\pi^{u}(\Sigma)I(\Sigma \in C_p)}{\pi^{u}(\Sigma \in C_p)}, ~ C_p = \{\mbox{ all } p \times p \mbox{ positive definite matrices }\}, $$ where $\pi^{u}(\cdot)$ is the unconstrained prior given by $$ \pi^{u}(\sigma_{jk} \mid \rho_{jk}) = N\left(\sigma_{jk} \mid 0, \frac{\rho_{jk}}{1 - \rho_{jk}}\tau_1^2\right) $$ $$ \pi^{u}(\rho_{jk}) = Beta(\rho_{jk} \mid a, b), ~ \rho_{jk} = 1 - 1/(1 + \phi_{jk}) $$ $$ \pi^{u}(\sigma_{jj}) = Exp(\sigma_{jj} \mid \lambda). $$ For more details, see Lee, Jo and Lee (2022). ## Examples ```r set.seed(1) n <- 20 p <- 5 # generate a sparse covariance matrix: True.Sigma <- matrix(0, nrow = p, ncol = p) diag(True.Sigma) <- 1 Values <- -runif(n = p*(p-1)/2, min = 0.2, max = 0.8) nonzeroIND <- which(rbinom(n=p*(p-1)/2,1,prob=1/p)==1) zeroIND = (1:(p*(p-1)/2))[-nonzeroIND] Values[zeroIND] <- 0 True.Sigma[lower.tri(True.Sigma)] <- Values True.Sigma[upper.tri(True.Sigma)] <- t(True.Sigma)[upper.tri(True.Sigma)] if(min(eigen(True.Sigma)$values) <= 0){ delta <- -min(eigen(True.Sigma)$values) + 1.0e-5 True.Sigma <- True.Sigma + delta*diag(p) } # generate a data X <- MASS::mvrnorm(n = n, mu = rep(0, p), Sigma = True.Sigma) # compute sparse, positive covariance estimator: fout <- bspcov::bmspcov(X = X, Sigma = diag(diag(cov(X)))) post.est.m <- bspcov::estimate(fout) sqrt(mean((post.est.m - True.Sigma)^2)) sqrt(mean((cov(X) - True.Sigma)^2)) ``` ## References Lee, K., Jo, S., and Lee, J. (2022), "The beta-mixture shrinkage prior for sparse covariances with near-minimax posterior convergence rate", **Journal of Multivariate Analysis**, 192, 105067. ## See Also sbmspcov ## Author(s) Kyoungjae Lee and Seongil Jo

bmspcov function