Generating positive-definite correlation matrices with additional structure

Many applied problems in the biological and social sciences involve estimating correlation matrices with special constraints. A consequence of these constraints is that the correlations in the random effects exhibit special symmetries, like block structure and/or Toeplitz-like diagonal-constant banding. Generating valid, positive-definite prior correlation matrices with such structure is typically non-trivial in the context of Bayesian model fitting. Here, we provide two effective solutions to the problem, the first based on an l2-norm penalty, and the second based on a flexible Cholesky factor parameterization that permits a priori constraints on the elements of the correlation matrix. We compare and contrast the methods using empirical and simulated network data, test for differences in efficiency, and highlight the partial strengths of each method, before discussing opportunities for further improvements. We also provide generalizable implementations of both methods in Stan code.