When, why, and how do diffusion posterior samplers fail? A finite-sample lens

Problem
This preprint addresses a significant gap in understanding the limitations of diffusion models used for posterior sampling in imaging inverse problems. While diffusion models are effective in modeling complex distributions, the reliance on inexact likelihood approximations at intermediate timesteps can lead to erroneous posterior distributions. The authors highlight that the empirical success of these approximations does not guarantee their theoretical soundness, leading to unexplained failures in posterior sampling. This work aims to elucidate the conditions under which these failures occur, providing a finite-sample perspective that has been largely overlooked in the literature.

Method
The authors introduce a finite-sample framework for posterior sampling that allows for the approximation of the posterior to arbitrary precision as the training set size approaches infinity. This framework is agnostic to the specific likelihood approximation and can accommodate any forward model and prior distribution. The key technical contribution lies in the analysis of how popular posterior sampling approximations can misrepresent the spread of the posterior at intermediate timesteps. The authors demonstrate that these inaccuracies can lead to various downstream issues, such as sensitivity to early stopping times and hallucination of modes not supported by the prior. The method serves as a diagnostic tool to evaluate existing and future posterior samplers, providing insights into their accuracy and potential failure modes.

Results
The authors empirically validate their finite-sample perspective by analyzing common posterior sampling approximations. They find that these approximations frequently under- or over-estimate the posterior spread, which can lead to significant errors in the sampled distributions. Although specific numerical results are not detailed in the abstract, the authors indicate that their findings reveal critical insights into the behavior of posterior samplers under various conditions, including the effects of multimodal priors and inaccurate intermediate likelihoods. The implications of these results suggest that many existing methods may not perform as expected in practical applications.

Limitations
The authors acknowledge that their finite-sample approach does not account for all potential sources of error in posterior sampling, particularly those arising from nonlinear measurement models. Additionally, while the framework is versatile, it may not capture all nuances of specific applications or model types. The authors do not address the computational overhead that may arise from implementing their diagnostic framework in practice, which could limit its applicability in real-time scenarios.

Why it matters
This work has significant implications for the development and evaluation of posterior samplers in machine learning and statistics. By providing a clearer understanding of the limitations of diffusion models and their likelihood approximations, the authors pave the way for more robust sampling techniques that can better handle complex distributions. The finite-sample perspective can inform future research on improving posterior sampling methods, potentially leading to advancements in fields such as Bayesian inference and generative modeling. This research encourages a reevaluation of existing methodologies and highlights the need for rigorous diagnostics in the development of new sampling algorithms.

Authors: Benjamin A. Burns, Sara Fridovich-Keil
Source: arXiv:2605.30330
URL: https://arxiv.org/abs/2605.30330v1