  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.
