Blade: A Derivative-free Bayesian Inversion Method using Diffusion Priors

TL;DR

Blade tackles derivative-free Bayesian inversion for black-box forward models by coupling a likelihood step based on statistical linearization with a diffusion-prior step implemented via denoising diffusion models, all within a Split Gibbs framework and an interacting-particle ensemble. It provides non-asymptotic convergence guarantees that quantify the impact of forward-model linearization and score approximation errors, and demonstrates superior posterior calibration and uncertainty quantification on Gaussian, Navier–Stokes, and image-related tasks. The method preserves multi-modality and uncertainty spread while remaining derivative-free, offering a practical pathway for uncertain inference in high-dimensional, nonlinear systems. The results suggest strong potential for real-world applications such as weather data assimilation and other domains requiring reliable uncertainty quantification under black-box forward models.

Abstract

Derivative-free Bayesian inversion is an important task in many science and engineering applications, particularly when computing the forward model derivative is computationally and practically challenging. In this paper, we introduce Blade, which can produce accurate and well-calibrated posteriors for Bayesian inversion using an ensemble of interacting particles. Blade leverages powerful data-driven priors based on diffusion models, and can handle nonlinear forward models that permit only black-box access (i.e., derivative-free). Theoretically, we establish a non-asymptotic convergence analysis to characterize the effects of forward model and prior estimation errors. Empirically, Blade achieves superior performance compared to existing derivative-free Bayesian inversion methods on various inverse problems, including challenging highly nonlinear fluid dynamics.

Figures (15)

• Figure 1: (a): Results on linear Gaussian and Gaussian mixture problems. Blue samples are from the ground-truth posterior. (b): Posterior draws from different methods on Navier-Stokes problem. “Observed GT” marks a single observed ground truth. Blade produces smooth, structured samples with realistic variability, while the competing methods yield noisier samples that stuck in a single blurred mode. See detailed comparison in Fig. \ref{['fig:ns-qualitative']}. (c): CRPS (continuous ranked probability score) versus SSR (spread-skill ratio) under varying measurement noise levels, with area indicating the relative runtime cost. Only Blade produces well-calibrated samples among derivative-free methods.
• Figure 2: Illustrative depiction of Blade (see Sec. \ref{['sec:method']}).
• Figure 3: Evolution of the rank of the space spanned by ensemble particles during Blade iterations.
• Figure 4: (a): Effect of different hyperparameters. $\gamma$: discretization step scale; $\rho_{\mathrm{min}}$: the minimum coupling strength. $\Tilde{\sigma}_{{\boldsymbol{y}}}$: the likelihood–spread factor. (b): Test-time scaling of Blade across different measurement noise levels. With more split Gibbs iterations, Blade not only becomes more accurate but also provides a more reliable assessment of its uncertainty.
• Figure 5: Illustration of the three annealing schedules. Each curve visualizes how the coupling strength $\rho_k$ evolves over iterations.

Theorems & Definitions (18)

• Remark 1
• Theorem 1: Stationary distribution
• Theorem 2: Convergence analysis
• Remark 2
• Remark 3
• Lemma 1: Stationary distribution of the likelihood step
• Lemma 1: Stationary distribution of the likelihood step
• proof
• Theorem 2: Stationary distribution
• proof