Figure 1: Comparison of network architectures (left) and flow trajectories (right). Discrete flows (NPE, top) require a specialized architecture for the density estimator. Continuous flows (FMPE, bottom) are based on a vector field parametrized with an unconstrained architecture.

Figure 1: Comparison of network architectures (left) and flow trajectories (right). Discrete flows (NPE, top) require a specialized architecture for the density estimator. Continuous flows (FMPE, bottom) are based on a vector field parametrized with an unconstrained architecture.

Figure 1: Comparison of network architectures (left) and flow trajectories (right). Discrete flows (NPE, top) require a specialized architecture for the density estimator. Continuous flows (FMPE, bottom) are based on a vector field parametrized with an unconstrained architecture.

Figure 1: Comparison of network architectures (left) and flow trajectories (right). Discrete flows (NPE, top) require a specialized architecture for the density estimator. Continuous flows (FMPE, bottom) are based on a vector field parametrized with an unconstrained architecture.

Amortized Bayesian model comparison (BMC) enables fast probabilistic ranking of models via simulation-based training of neural surrogates. However, the reliability of neural surrogates deteriorates when simulation models are misspecified - the very case where model comparison is most needed. Thus, we supplement simulation-based training with a self-consistency (SC) loss on unlabeled real data to improve BMC estimates under empirical distribution shifts. Using a numerical experiment and two case studies with real data, we compare amortized evidence estimates with and without SC against analytic or bridge sampling benchmarks. SC improves calibration under model misspecification when having access to analytic likelihoods. However, it offers limited gains with neural surrogate likelihoods, making it most practical for trustworthy BMC when likelihoods are exact.

A central challenge in many areas of science and engineering is to identify model parameters that are consistent with prior knowledge and empirical data. Bayesian inference offers a principled framework for this task, but can be computationally prohibitive when models are defined by stochastic simulators. Simulation-based Inference (SBI) is a suite of methods developed to overcome this limitation, which has enabled scientific discoveries in fields such as particle physics, astrophysics, and neuroscience. The core idea of SBI is to train neural networks on data generated by a simulator, without requiring access to likelihood evaluations. Once trained, inference is amortized: The neural network can rapidly perform Bayesian inference on empirical observations without requiring additional training or simulations. In this tutorial, we provide a practical guide for practitioners aiming to apply SBI methods. We outline a structured SBI workflow and offer practical guidelines and diagnostic tools for every stage of the process -- from setting up the simulator and prior, choosing and training inference networks, to performing inference and validating the results. We illustrate these steps through examples from astrophysics, psychophysics, and neuroscience. This tutorial empowers researchers to apply state-of-the-art SBI methods, facilitating efficient parameter inference for scientific discovery.

Neural simulation-based inference (SBI) is a popular set of methods for Bayesian inference when models are only available in the form of a simulator. These methods are widely used in the sciences and engineering, where writing down a likelihood can be significantly more challenging than constructing a simulator. However, the performance of neural SBI can suffer when simulators are computationally expensive, thereby limiting the number of simulations that can be performed. In this paper, we propose a novel approach to neural SBI which leverages multilevel Monte Carlo techniques for settings where several simulators of varying cost and fidelity are available. We demonstrate through both theoretical analysis and extensive experiments that our method can significantly enhance the accuracy of SBI methods given a fixed computational budget.