The Statistical Accuracy of Neural Posterior and Likelihood Estimation

These methods can approximate the likelihood through neural likelihood estimation (NLE) (Papamakarios et al., 2019) or directly target the posterior distribution with neural posterior estimation (NPE) (Greenberg et al., 2019; Lueckmann et al., 2017; Papamakarios and Murray, 2016), with NLE requiring subsequent Markov Chain Monte Carlo (MCMC) steps to produce posterior samples. The hallmark of these neural methods is their ability to accurately approximate complex posterior distributions using only forward simulations from the assumed model. While sequential methods iteratively refine the posterior estimate through multiple rounds of simulation, one-shot NPE and NLE methods perform inference in a single round, enabling amortized inference where a trained model can be reused for multiple datasets without retraining (see, e.g., Radev et al., 2020; Gloeckler et al., 2024). In particular, like the statistical methods of approximate Bayesian computation (ABC), see, e.g., Sisson et al. (2018) for a handbook treatment, and Martin et al. (2023) for a recent summary, and Bayesian synthetic likelihood (BSL), see, e.g., Wood (2010), Price et al. (2018) and Frazier et al. (2023), NPE and NLE first reduce the data down to a vector of statistics and then build an approximation to the resulting partial posterior by substituting likelihood evaluation with forward simulation from the assumed model. In contrast to the statistical methods for likelihood-free inference like ABC and BSL, NPE (respectively, NLE) approximates the posterior (resp., the likelihood) directly by fitting flexible conditional density estimators, usually neural-or flow-based approaches, using training data that is simulated from the assumed model space. The approximation that results from this training step is then directly used as a posterior in the context of NPE or as a likelihood in the case of NLE, with MCMC for this trained likelihood then used to produce draws from an approximate posterior.