Bayesian shrinkage wavelet estimation of mean matrix of the matrix variate normal distribution with application in de-noising

Suppose that the random matrix \({\textbf{X}}_{p\times m}\) ${\mathbf{X}}_{p\times m}$ has a matrix variate normal distribution with the mean matrix \(\varvec{\Theta }\) $\mathbf{\Theta }$ and covariance matrix \({{\varvec{\Sigma }}} \otimes {\varvec{\Psi }}\) $\mathbf{\Sigma }\otimes \mathbf{\Psi }$ , where \({{\varvec{\Sigma }}}\) $\mathbf{\Sigma }$ and \({\varvec{\Psi }}\) $\mathbf{\Psi }$ are known positive definite covariance matrices. This paper studies the Bayesian shrinkage wavelet estimation of the mean matrix \(\varvec{\Theta }\) $\mathbf{\Theta }$ under the balanced loss function. Two soft Bayesian shrinkage wavelet estimators are proposed based on two prior distributions: the matrix variate normal \(N_{p,m}({\textbf{0}}, {\varvec{\Lambda }}\otimes {\varvec{\Psi }})\) $N_{p,m}(\mathbf{0},\mathbf{\Lambda }\otimes \mathbf{\Psi })$ , where \({\varvec{\Lambda }}\) $\mathbf{\Lambda }$ is a known positive definite covariance matrix, and the improper prior \(\pi (\varvec{\Theta })=1\) $\pi (\mathbf{\Theta })=1$ . Using Bayes estimators as the target estimator and Stein’s unbiased risk estimate technique, the soft Bayesian shrinkage wavelet threshold and the soft generalized Bayesian shrinkage wavelet threshold are obtained. Based on the newly proposed thresholds, we derive the soft Bayesian shrinkage wavelet and the soft generalized Bayesian shrinkage wavelet estimators. The performance of the presented theoretical topics is measured through a simulation study and three real examples. The results show that the soft generalized Bayesian shrinkage wavelet estimator outperforms four classical soft shrinkage wavelet estimators and the soft Bayesian shrinkage wavelet estimator.