Generating large Ising models with Markov structure via simple linear relations

We extend the notion of a tree graph to sequences of prime graphs which are cycles and edges and name these non-chordal graphs hollow trees. These structures are especially attractive for palindromic Ising models, which mimic a symmetry of joint Gaussian distributions. We show that for an Ising model all defining independences are captured by zero partial correlations and conditional correlations agree with partial correlations within each prime graph if and only if the model is palindromic and has a hollow-tree structure. This implies that the strength of dependences can be assessed locally. We use the results to find a well-fitting general Ising model with hollow-tree structure for a set of longitudinal data.