  Diffusion models are a class of probabilistic generative models that have
been widely used as a prior for image processing tasks like text conditional
generation and inpainting. We demonstrate that these models can be adapted to
make predictions and provide uncertainty quantification for chaotic dynamical
systems. In these applications, diffusion models can implicitly represent
knowledge about outliers and extreme events; however, querying that knowledge
through conditional sampling or measuring probabilities is surprisingly
difficult. Existing methods for conditional sampling at inference time seek
mainly to enforce the constraints, which is insufficient to match the
statistics of the distribution or compute the probability of the chosen events.
To achieve these ends, optimally one would use the conditional score function,
but its computation is typically intractable. In this work, we develop a
probabilistic approximation scheme for the conditional score function which
provably converges to the true distribution as the noise level decreases. With
this scheme we are able to sample conditionally on nonlinear userdefined events
at inference time, and matches data statistics even when sampling from the
tails of the distribution.
