Kalman Posterior Sampling with Diffusion Priors and Simulation-Based Likelihoods

In science, measurements are often modeled by stochastic numerical simulators whose randomness is non-Gaussian and whose likelihood and gradients are rarely available. Diffusion posterior sampling usually assumes all three, which rules these simulators out. We introduce Kalman Posterior Sampling (KPS), a sampler in which the diffusion prior and a statistical linearization of the likelihood defined by the simulator enter a common Kalman update through their moments, using simulations alone or gradients when the simulator is differentiable. An inner Gibbs loop keeps the iterates consistent with the diffusion process, preventing errors from accumulating across steps. On latent fluid dynamics and scientific inverse problem benchmarks, KPS outperforms state-of-the-art samplers in accuracy with as few as 16 particles, delivers close-to-calibrated uncertainty, and captures multimodal posteriors even from black-box simulators.