In the application of the Expectation Maximization (EM) algorithm to identification of dynamical systems, latent variables are typically taken as system states, for simplicity. In this work, we propose a different choice of latent variables, namely, system disturbances. Such a formulation is shown, under certain circumstances, to improve the fidelity of bounds on the likelihood, and circumvent difficulties related to intractable model transition densities. To access these benefits, we propose a Lagrangian relaxation of the challenging optimization problem that arises when formulating over latent disturbances, and fully develop the method for linear models.