  Score-based diffusion models (SDMs) offer a flexible approach to sample from
the posterior distribution in a variety of Bayesian inverse problems. In the
literature, the prior score is utilized to sample from the posterior by
different methods that require multiple evaluations of the forward mapping in
order to generate a single posterior sample. These methods are often designed
with the objective of enabling the direct use of the unconditional prior score
and, therefore, task-independent training. In this paper, we focus on linear
inverse problems, when evaluation of the forward mapping is computationally
expensive and frequent posterior sampling is required for new measurement data,
such as in medical imaging. We demonstrate that the evaluation of the forward
mapping can be entirely bypassed during posterior sample generation. Instead,
without introducing any error, the computational effort can be shifted to an
offline task of training the score of a specific diffusion-like random process.
In particular, the training is task-dependent requiring information about the
forward mapping but not about the measurement data. It is shown that the
conditional score corresponding to the posterior can be obtained from the
auxiliary score by suitable affine transformations. We prove that this
observation generalizes to the framework of infinite-dimensional diffusion
models introduced recently and provide numerical analysis of the method.
Moreover, we validate our findings with numerical experiments.
