## Appendix D Alternative Approach for Modeling \(\mathbf{\Sigma}\)  
Although the Wishart is a standard prior for the covariance matrix and convenient to use, it has been criticized for being too restricted and the lack of flexibility. Also, as pointed out by [PERSON] _et al._[52] and [PERSON] [50], if the true covariance matrix \(\mathbf{\Sigma}\) is close to \(I\), the eigenstructure of \(\mathbf{\Sigma}\) can be systematically distorted by the estimator, so this conventional prior can behave poorly, especially when the sample size is small or the data are sparse. To overcome these drawbacks, several other more flexible priors have been introduced, including a log-matrix prior [51], a reference noninformative prior [53], and a constrained Wishart prior [54]. The covariance matrix can also be modeled with a different parameterization [55]. This strategy is based on the key idea that a covariance matrix for longitudinal data can be diagonalized, i.e.,  
\[\mathbf{A}^{\prime}\mathbf{\Sigma}\mathbf{A}=\mathbf{S}, \tag{11}\]  
where \(\mathbf{S}\) is a diagonal matrix with positive entries and \(\mathbf{A}\) is a unique lower triangular matrix with 1's on the diagonal. Without nice properties due to the conjugate priors, the resulting full conditional for \(\mathbf{\Sigma}\) is no longer inverse Wishart. Often, one might have to generate \(\mathbf{\Sigma}\) componentwise by using Gibbs sampling. Consequently, we will focus on computationally easy methods that can generate the entire \(\mathbf{\Sigma}\) at a time, given the problem's complexity. Therefore, as an option to further improve the estimation for the covariance matrix, the reference (noninformative) prior, first introduced by [PERSON] and [PERSON] [56] and thoroughly discussed by [PERSON] and [PERSON] [53], is investigated. The Jeffreys prior,  
\[\pi(\mathbf{\Sigma})=|\mathbf{\Sigma}|^{-(p+1)/2} \tag{12}\]  
might be the most commonly used reference prior. However, care must be taken when using this prior, as it can lead to an improper posterior distribution, and it also fails to shrink the eigenvalues appropriately. However, the approach proposed by [PERSON] and [PERSON] [53] has proven to be able to overcome these inadequacies of the Jeffrey's prior remarkably. Note that \(\mathbf{\Sigma}\) can be decomposed as \(\mathbf{\Sigma}=\mathbf{ODO}^{\prime}\), where \(\mathbf{O}\) is orthogonal with positive entries in the first row, and \(\mathbf{D}=diag(d_{1},d_{2},...,d_{p})\), with \(d_{1}\geqslant,d_{2}\geqslant,...,\geqslant d_{p}\). Hence, providing the monotonically ordered \(\{d_{i}\}\), the reference prior for \((\mathbf{D},\mathbf{O})\) is given by:  
\[\pi(\mathbf{D},\mathbf{O})\propto\frac{1}{|\mathbf{\Sigma}|\prod_{i<j}(d_{i}-d_{j })}\;, \tag{13}\]and the resulting posterior distribution is:  
\[\pi(\mathbf{\Sigma}|\mathbf{y},\mathbf{\Theta},\mathbf{Q},\lambda)\propto\frac{ \exp[tr(-\frac{1}{2n}\mathbf{\Theta}\mathbf{D}^{-1}\mathbf{O}^{\prime}(\sum_{j=0 }^{2}\sum_{i=1}^{n_{j}}(\mathbf{y}_{ij})-g(\mathbf{t}|\mathbf{\Theta}_{j}))( \mathbf{y}_{ij})-g(\mathbf{t}|\mathbf{\Theta}_{j}))^{\prime}))]}{|\mathbf{ \Sigma}|^{n/2+1}}. \tag{10}\]  
Comparing the reference prior with the [PERSON]'s prior, it is noted that the posterior given in equation (10) is always proper. Also note that, since this reference prior put more mass near the region for equal eigenvalues, it can produce an estimator with better eigenstructure. Again, it is very difficult to analytically evaluate the posterior (10). [PERSON] and [PERSON] [53] suggested using a Metroplisized hit-and-run sampler algorithm to obtain the integration. The detail sampling procedure at the \(kth\) iteration is given as follows:  
1. Given the current positive-definitive matrix \(\mathbf{\Sigma}_{k}\), we set \(\mathbf{W}_{k}=\log\mathbf{\Sigma}_{k}\), in the sense that \(\mathbf{W}_{k}=\sum_{i=0}^{\infty}\frac{(\mathbf{\Sigma}_{k})^{i}}{i! }\);
2.