EM vs MM: A case study

The celebrated expectation-maximization (EM) algorithm is one of the most widely used optimization methods in statistics. In recent years it has been realized that EM algorithm is a special case of the more general minorization-maximization (MM) principle. Both algorithms creates a surrogate function in the first (E or M) step that is maximized in the second M step. This two step process always drives the objective function uphill and is iterated until the parameters converge. The two algorithms differ in the way the surrogate function is constructed. The expectation step of the EM algorithm relies on calculating conditional expectations, while the minorization step of the MM algorithm builds on crafty use of inequalities. For many problems, EM and MM derivations yield the same algorithm. This expository note walks through the construction of both algorithms for estimating the parameters of the Dirichlet-Multinomial distribution. This particular case is of interest because EM and MM derivations lead to two different algorithms with completely distinct operating characteristics. The EM algorithm converges fast but involves solving a nontrivial maximization problem in the M step. In contrast the MM updates are extremely simple but converge slowly. An EM-MM hybrid algorithm is derived which shows faster convergence than the MM algorithm in certain parameter regimes. The local convergence rates of the three algorithms are studied theoretically from the unifying MM point of view and also compared on numerical examples.