Mixture prior for sparse signals with dependent covariance structure

In this study, we propose an estimation method for normal mean problem that can have unknown sparsity as well as correlations in the signals. Our proposed method first decomposes arbitrary dependent covariance matrix of the observed signals into two parts: common dependence and weakly dependent error terms. By subtracting common dependence, the correlations among the signals are significantly weakened. It is practical for doing this because of the existence of sparsity. Then the sparsity is estimated using an empirical Bayesian method based on the likelihood of the signals with the common dependence removed. Using simulated examples that have moderate to high degrees of sparsity and different dependent structures in the signals, we demonstrate that the performance of our proposed algorithm is favorable compared to the existing method which assumes the signals are independent identically distributed. Furthermore, our approach is applied on the widely used "Hapmap" gene expressions data, and our results are consistent with the findings in other studies.