Data are assumed to originate from a design with replicates. Each observation \(\mathbf{Y}_{i,k_i}\) with \(k_i\) (\(k_i = 1, \ldots, K_i\)) the \(k_i\)-th replicate of the \(i\)-th sample, is described by a `signal+noise' model: \(\mathbf{Y}_{i,k_i} = \mathbf{Z}_i + \boldsymbol{\varepsilon}_{i,k_i}\), where \(\mathbf{Z}_i\) and \(\boldsymbol{\varepsilon}_{i,k_i}\) represent the signal and noise, respectively. Each observation \(\mathbf{Y}_{i,k_i}\) follows a multivariate normal law of the form
\(\mathbf{Y}_{i,k_i} \sim \mathcal{N}(\mathbf{0}_p, \boldsymbol{\Omega}_z^{-1} + \boldsymbol{\Omega}_{\varepsilon}^{-1})\), which results from the distributional assumptions of the signal and the noise, \(\mathbf{Z}_{i} \sim \mathcal{N}(\mathbf{0}_p, \boldsymbol{\Omega}_z^{-1})\) and \(\boldsymbol{\varepsilon}_{i, k_i} \sim \mathcal{N}(\mathbf{0}_p, \boldsymbol{\Omega}_{\varepsilon}^{-1})\), and their independence. The model parameters are estimated by means of a penalized EM algorithm that maximizes the loglikelihood augmented with the penalty \(\lambda_z \| \boldsymbol{\Omega}_z - \mathbf{T}_z \|_F^2 + \lambda_{\varepsilon} \| \boldsymbol{\Omega}_{\varepsilon} - \mathbf{T}_{\varepsilon} \|_F^2\), in which \(\mathbf{T}_z\) and \(\mathbf{T}_{\varepsilon}\) are the shrinkage targets of the signal and noise precision matrices, respectively. For more details see van Wieringen and Chen (2019).