Likelihood inference with hierarchical models is often complicated by the fact that the likelihood function involves intractable integrals. Numerical integration (e.g. quadrature) is an option if the dimension of the integral is low but quickly becomes unreliable as the dimension grows. An alternative approach is to approximate the intractable integrals using Monte Carlo averages. Several dierent algorithms based on this idea have been proposed. In this paper we discuss the relative merits of simulated maximum likelihood, Monte Carlo EM, Monte Carlo Newton-Raphson and stochastic approximation. Key words and phrases : Eciency, Monte Carlo EM, Monte Carlo Newton-Raphson, Rate of convergence, Simulated maximum likelihood, Stochastic approximation All three authors partially supported by NSF Grant DMS-00-72827. 1 1