Title: Maximum Likelihood Learning of Energy-Based Models for Simulation-Based Inference. Abstract: We introduce two Synthetic Likelihood methods for Simulation-Based Inference (SBI), to conduct either amortized or targeted inference from experimental observations when a high-fidelity simulator is available. Both methods learn a Conditional Energy-Based Model (EBM) of the likelihood using synthetic data generated by the simulator, conditioned on parameters drawn from a proposal distribution. The learned likelihood can then be combined with any prior to obtain a posterior estimate, from which samples can be drawn using MCMC. 
Our methods uniquely combine a flexible Energy-Based Model and the minimization of a KL loss: this is in contrast to other synthetic likelihood methods, which either rely on normalizing flows, or minimize score-based objectives; choices that come with known pitfalls. Our first method, Amortized Unnormalized Neural Likelihood Estimation (AUNLE), introduces a tilting trick during training that allows to perform inference using efficient MCMC techniques. Our second method, Sequential UNLE (SUNLE), employs a doubly intractable approach in order to re-use simulation data and improve posterior accuracy for a specific observation. 
We demonstrate the properties of both methods on a range of synthetic datasets, and apply it to a neuroscience model of the pyloric network in the crab, matching the performance of other synthetic likelihood methods at a fraction of the simulation budget.