Beyond estimating parameters of interest from data, one of the key goals of statistical inference is to properly quantify uncertainty in these estimates. In Bayesian inference, this uncertainty is provided by the posterior distribution, the computation of which typically involves an intractable high-dimensional integral. Among available approximation methods, sampling-based approaches come with strong theoretical guarantees but scale poorly to large problems, while variational approaches scale well but offer few theoretical guarantees. In particular, variational methods are known to produce overconfident estimates of posterior uncertainty and are typically non-identifiable, with many latent variable configurations generating equivalent predictions. Here, we address these challenges by showing how diffusion-based models (DBMs), which have recently produced state-of-the-art performance in generative modeling tasks, can be repurposed for performing calibrated, identifiable Bayesian inference. By exploiting a previously established connection between the stochastic and probability flow ordinary differential equations (pfODEs) underlying DBMs, we derive a class of models, inflationary flows, that uniquely and deterministically map high-dimensional data to a lower-dimensional Gaussian distribution via ODE integration. This map is both invertible and neighborhood-preserving, with controllable numerical error, with the result that uncertainties in the data are correctly propagated to the latent space. We demonstrate how such maps can be learned via standard DBM training using a novel noise schedule and are effective at both preserving and reducing intrinsic data dimensionality. The result is a class of highly expressive generative models, uniquely defined on a low-dimensional latent space, that afford principled Bayesian inference.

翻译：除了从数据中估计感兴趣的参数外，统计推断的一个关键目标是正确量化这些估计的不确定性。在贝叶斯推断中，这种不确定性由后验分布提供，其计算通常涉及难以处理的高维积分。在现有的近似方法中，基于采样的方法具有坚实的理论保证，但难以扩展到大规模问题；而变分方法扩展性良好，但缺乏理论保证。特别是，变分方法已知会产生过于自信的后验不确定性估计，且通常不可识别——许多潜变量配置能产生等价的预测。本文通过展示如何将近期在生成建模任务中取得最先进性能的扩散模型重新用于执行校准的、可识别的贝叶斯推断，以应对这些挑战。通过利用先前建立的、扩散模型基础中的随机微分方程与概率流常微分方程之间的关联，我们推导出一类称为膨胀流的模型，其通过ODE积分将高维数据唯一且确定性地映射到低维高斯分布。该映射既可逆又保持邻域关系，且具有可控的数值误差，从而确保数据中的不确定性被正确传播到潜空间。我们证明了此类映射可通过采用新颖噪声调度的标准扩散模型训练来学习，并能有效保持和降低数据的内在维度。其结果是得到一类高度表达性的生成模型，这些模型在低维潜空间上唯一定义，并支持原则性的贝叶斯推断。