Bayesian inference for complex models with an intractable likelihood can be tackled using algorithms performing many calls to computer simulators. These approaches are collectively known as "simulation-based inference" (SBI). Recent SBI methods have made use of neural networks (NN) to provide approximate, yet expressive constructs for the unavailable likelihood function and the posterior distribution. However, they do not generally achieve an optimal trade-off between accuracy and computational demand. In this work, we propose an alternative that provides both approximations to the likelihood and the posterior distribution, using structured mixtures of probability distributions. Our approach produces accurate posterior inference when compared to state-of-the-art NN-based SBI methods, while exhibiting a much smaller computational footprint. We illustrate our results on several benchmark models from the SBI literature.

State-of-the-art neural network-based methods for learning summary statistics have delivered promising results for simulation-based likelihood-free parameter inference. Existing approaches require density estimation as a post-processing step building upon deterministic neural networks, and do not take network prediction uncertainty into account. This work proposes a robust integrated approach that learns summary statistics using Bayesian neural networks, and directly estimates the posterior density using categorical distributions. An adaptive sampling scheme selects simulation locations to efficiently and iteratively refine the predictive posterior of the network conditioned on observations. This allows for more efficient and robust convergence on comparatively large prior spaces. We demonstrate our approach on benchmark examples and compare against related methods.

Likelihood-free methods perform parameter inference in stochastic simulator models where evaluating the likelihood is intractable but sampling synthetic data is possible. One class of methods for this likelihood-free problem uses a classifier to distinguish between pairs of parameter-observation samples generated using the simulator and pairs sampled from some reference distribution, which implicitly learns a density ratio proportional to the likelihood. Another popular class of methods fits a conditional distribution to the parameter posterior directly, and a particular recent variant allows for the use of flexible neural density estimators for this task. In this work, we show that both of these approaches can be unified under a general contrastive learning scheme, and clarify how they should be run and compared.