  Denoisers play a central role in many applications, from noise suppression in
low-grade imaging sensors, to empowering score-based generative models. The
latter category of methods makes use of Tweedie's formula, which links the
posterior mean in Gaussian denoising (\ie the minimum MSE denoiser) with the
score of the data distribution. Here, we derive a fundamental relation between
the higher-order central moments of the posterior distribution, and the
higher-order derivatives of the posterior mean. We harness this result for
uncertainty quantification of pre-trained denoisers. Particularly, we show how
to efficiently compute the principal components of the posterior distribution
for any desired region of an image, as well as to approximate the full marginal
distribution along those (or any other) one-dimensional directions. Our method
is fast and memory-efficient, as it does not explicitly compute or store the
high-order moment tensors and it requires no training or fine tuning of the
denoiser. Code and examples are available on the project webpage in
https://hilamanor.github.io/GaussianDenoisingPosterior/ .
