The likelihood function is the basis of many statistical inference procedures. However, in various areas like population genetics, systems biology, epidemiology, queuing systems and spatial statistics, complex statistical models are required for which the likelihood cannot be obtained analytically. In recent years, increasing computing power has allowed to circumvent this problem by simulation-based methods like Indirect Inference and Approximate Bayesian Computation.

Here, we propose an alternative approach that is based on stochastic gradient methods. By moving along a simulated gradient, the algorithm produces a sequence of estimates that will eventually converge to the maximum likelihood estimate (or, equivalently, to the maximum of the posterior). This approach reduces the number of simulations in regions of low likelihood while being flexibly applicable to a large variety of problems.

We present a set of conditions under which the algorithm converges to the maximum likelihood estimate w. p. 1 and we also explore the properties of the resulting estimator in practical applications. To this end we first propose a set of tuning guidelines that improve the robustness of the algorithm against too noisy simulation results. Then, we investigate the performance of our approach in simulation studies and apply our algorithm to two models with intractable likelihood functions. First, we present an application in the context of queuing systems. Second, we re-analyse population genetic data and estimate parameters describing the demographic history of Bornean and Sumatran orangutan populations.