Variational upper and lower bounds for probabilistic graphical models

Probabilistic phylogenetic models which relax the site independence evolution assumption often face the problem of infeasible likelihood computations, for example, for the task of selecting suitable parameters for the model. We present a new approximation method, applicable for a wide range of probabilistic models, which guarantees to upper and lower bound the true likelihood of data, and apply it to the problem of probabilistic phylogenetic models. The new method is complementary to known variational methods that lower bound the likelihood, and it uses similar methods to optimize the bounds from above and below. We applied our method to aligned DNA sequences of various lengths from human in the region of the CFTR gene and homologous from eight mammals, and found the bounds to be appreciably close to the true likelihood whenever it could be computed. When computing the exact likelihood was not feasible, we demonstrated the proximity of the upper and lower variational bounds, implying a tight approximation of the likelihood.