  In this article we introduce a new family of lattice polytopes with rational
linear precision. For this purpose, we define a new class of discrete
statistical models that we call multinomial staged tree models. We prove that
these models have rational maximum likelihood estimators (MLE) and give a
criterion for these models to be log-linear. Our main result is then obtained
by applying Garcia-Puente and Sottile's theorem that establishes a
correspondence between polytopes with rational linear precision and log-linear
models with rational MLE. Throughout this article we also study the interplay
between the primitive collections of the normal fan of a polytope with rational
linear precision and the shape of the Horn matrix of its corresponding
statistical model. Finally, we investigate lattice polytopes arising from toric
multinomial staged tree models, in terms of the combinatorics of their tree
representations.
