  Diffusion generative models unlock new possibilities for inverse problems as
they allow for the incorporation of strong empirical priors into the process of
scientific inference. Recently, diffusion models received significant attention
for solving inverse problems by posterior sampling, but many challenges remain
open due to the intractability of this sampling process. Prior work resorted to
Gaussian approximations to conditional densities of the reverse process,
leveraging Tweedie's formula to parameterise its mean, complemented with
various heuristics. In this work, we leverage higher order information using
Tweedie's formula and obtain a finer approximation with a principled covariance
estimate. This novel approximation removes any time-dependent step-size
hyperparameters required by earlier methods, and enables higher quality
approximations of the posterior density which results in better samples.
Specifically, we tackle noisy linear inverse problems and obtain a novel
approximation to the gradient of the likelihood. We then plug this gradient
estimate into various diffusion models and show that this method is optimal for
a Gaussian data distribution. We illustrate the empirical effectiveness of our
approach for general linear inverse problems on toy synthetic examples as well
as image restoration using pretrained diffusion models as the prior. We show
that our method improves the sample quality by providing statistically
principled approximations to diffusion posterior sampling problem.
