Diffusion Models in Simulation-Based Inference: A Tutorial Review
Authors: Jonas Arruda, Niels Bracher, Ullrich Köthe, Jan Hasenauer, Stefan T. Radev
Abstract
Diffusion models have recently emerged as powerful learners for simulation-based inference (SBI), enabling fast and accurate estimation of latent parameters from simulated and real data. Their score-based formulation offers a flexible way to learn conditional or joint distributions over parameters and observations, thereby providing a versatile solution to various modeling problems. In this tutorial review, we synthesize recent developments on diffusion models for SBI, covering design choices for training, inference, and evaluation. We highlight opportunities created by various concepts such as guidance, score composition, flow matching, consistency models, and joint modeling. Furthermore, we discuss how efficiency and statistical accuracy are affected by noise schedules, parameterizations, and samplers. Finally, we illustrate these concepts with case studies across parameter dimensionalities, simulation budgets, and model types, and outline open questions for future research.
The performance of generative diffusion models is determined by the choice of the reference diffusion process connecting the empirical and prior distributions. Conventional approaches typically trade off simulation-free training against finite-time generation. We propose a framework for designing the reference process that achieves both simultaneously. The key idea is to prescribe tractable time-dependent conditional distributions and then construct the reference process realizing them as its marginals. This framework reveals that score matching is not fundamental to diffusion-model training but instead emerges naturally through reversal of the reference process. We further show that conditional flow matching arises as the small-noise limit of the proposed framework.
Flow and diffusion generative models have established themselves as widely adopted density estimators for simulation-based inference (SBI), extending naturally from neural posterior estimation to likelihood and joint density estimation. Their principled optimization objectives and freedom from architectural constraints have driven rapid adoption across the natural sciences. Yet the most widely used SBI libraries remain PyTorch-based, leaving researchers who develop their forward models and analysis pipelines in JAX without a native option. We present GenSBI, an open-source library that implements flow matching, score matching, and denoising diffusion entirely in JAX. The library offers three transformer-based architectures - SimFormer, Flux1, and a novel Flux1Joint that extends gate-modulated transformer blocks to joint density estimation - all interchangeable through a unified interface that decouples generative method, neural backbone, and inference mode. GenSBI provides an end-to-end workflow from training through posterior calibration (SBC, TARP, LC2ST) and supports custom architectures with domain-specific embedding networks. We validate the framework on standard SBI benchmarks, achieving near-ideal mean C2ST scores (0.50-0.56, where 0.50 is ideal) on SBIBM tasks with minimal per-task tuning and well-calibrated posterior coverage across all tested configurations. The code is publicly available at https://github.com/aurelio-amerio/GenSBI.
The cost of simulator evaluations is a key practical bottleneck for Simulation Based Inference (SBI). In hierarchical settings with shared global parameters and exchangeable site-level parameters and observations, this structure can be exploited to improve simulation efficiency. Existing hierarchical SBI approaches factorise the posterior yet still simulate across multiple sites per training sample; We instead explore likelihood factorisation (LF) to train from single-site simulations. In LF sampling we learn a per-site neural surrogate of the simulator and then assemble synthetic multi-site observations to amortise inference for the full hierarchical posterior. Building on this, we propose Tokenised Flow Matching for Posterior Estimation (TFMPE), a tokenised flow matching approach that supports function-valued observations through likelihood factorisation. To enable systematic evaluation, we introduce a benchmark for hierarchical SBI. We validate TFMPE on this benchmark and on realistic infectious disease and computational fluid dynamics models, finding well-calibrated posteriors while reducing computational cost.