Before revision:  To implement this multi-family EC analysis strategy, we use a set of  correlated multivariate Gaussian distributions to model the related families and then co-estimate the inverse covariance matrices of the Gaussian distributions subject to the constraint that these matrices shall have similar patternsto some degree.
After revision:  To implement this multi-family EC analysis strategy, we use a set of correlated multivariate Gaussian distributions , the inverse covariance matrix (or precision matrix) of each distribution encoding the contact pattern of one family. Then we co-estimate the inverse covariance matrices of the Gaussian distributions subject to the constraint that these matrices shall have similar patternsto some degree.