Diffusion Schr\"{o}dinger Bridges for Bayesian Computation

Abstract

Denoising diffusion models are a novel class of generative models that have recently become extremely popular in machine learning. In this paper, we describe how such ideas can also be used to sample from posterior distributions and, more generally, any target distribution whose density is known up to a normalizing constant. The key idea is to consider a forward ``noising'' diffusion initialized at the target distribution which ``transports'' this latter to a normal distribution for long diffusion times. The time-reversal of this process, the ``denoising'' diffusion, thus ``transports'' the normal distribution to the target distribution and can be approximated so as to sample from the target. To accelerate simulation, we show how one can introduce and approximate a Schr\"{o}dinger bridge between these two distributions, i.e. a diffusion which transports the normal to the target in finite time.