A Review of Diffusion-based Simulation-Based Inference: Foundations and Applications in Non-Ideal Data Scenarios

Abstract: For complex simulation problems, inferring parameters of scientific interest often precludes the use of classical likelihood-based techniques due to intractable likelihood functions. Simulation-based inference (SBI) methods forego the need for explicit likelihoods by directly utilizing samples from the simulator to learn posterior distributions over parameters $\mathbfθ$ given observed data $\mathbf{x}_{\text{o}}$. Recent work has brought attention to diffusion models -- a type of generative model rooted in score matching and reverse-time stochastic dynamics -- as a flexible framework SBI tasks. This article reviews diffusion-based SBI from first principles to applications in practice. We first recall the mathematical foundations of diffusion modeling (forward noising, reverse-time SDE/ODE, probability flow, and denoising score matching) and explain how conditional scores enable likelihood-free posterior sampling. We then examine where diffusion models address pain points of normalizing flows in neural posterior/likelihood estimation and where they introduce new trade-offs (e.g., iterative sampling costs). The key theme of this review is robustness of diffusion-based SBI in non-ideal conditions common to scientific data: misspecification (mismatch between simulated training data and reality), unstructured or infinite-dimensional observations, and missingness. We synthesize methods spanning foundations drawing from Schrodinger-bridge formulations, conditional and sequential posterior samplers, amortized architectures for unstructured data, and inference-time prior adaptation. Throughout, we adopt consistent notation and emphasize conditions and caveats required for accurate posteriors. The review closes with a discussion of open problems with an eye toward applications of uncertainty quantification for probabilistic geophysical models that may benefit from diffusion-based SBI.