  In this work we investigate partition models, the subset of log-linear models
for which one can perform the iterative proportional scaling (IPS) algorithm to
numerically compute the maximum likelihood estimate (MLE). Partition models
include families of models such as hierarchical models and balanced, stratified
staged trees. We define a sufficient condition, called the Generalized Running
Intersection Property (GRIP), on the matrix representation of a partition model
under which IPS algorithm produces the exact MLE in one cycle. Additionally we
connect the GRIP to the toric fiber product and to previous results for
hierarchical models and balanced, stratified staged trees. This leads to a
characterization of balanced, stratified staged trees in terms of the GRIP.
