Covariance Partition Priors: A Bayesian Approach to Simultaneous Covariance Estimation for Longitudinal Data

When data consists of multiple groups, it is often assumed the covariance matrices are either equal across groups or are completely distinct. We seek methodology to allow borrowing of strength across potentially similar groups to improve estimation.

To that end, we introduce a covariance partition prior which proposes a partition of the groups at each measurement time. Groups in the same set of the partition share dependence parameters for the distribution of the current measurement given the preceding ones, and the sequence of partitions is modeled as a Markov chain to encourage similar structure at nearby measurement times. This approach additionally encourages a lower-dimensional structure of the covariance matrices by shrinking the parameters of the Cholesky decomposition toward zero.