Modern datasets are often in the form of matrices or arrays, potentially
having correlations along each set of data indices. For example, data involving re-
peated measurements of several variables over time may exhibit temporal correla-
tion as well as correlation among the variables. A possible model for matrix-valued
data is the class of matrix normal distributions, which is parametrized by two co-
variance matrices, one for each index set of the data. In this article we discuss
an extension of the matrix normal model to accommodate multidimensional data
arrays, or tensors. We show how a particular array-matrix product can be used
to generate the class of array normal distributions having separable covariance
structure. We derive some properties of these covariance structures and the cor-
responding array normal distributions, and show how the array-matrix product
can be used to de¯ne a semi-conjugate prior distribution and calculate the corre-
sponding posterior distribution. We illustrate the methodology in an analysis of
multivariate longitudinal network data which take the form of a four-way arra