BIGMC: where Bayesian inverse problems, generative models, and Monte Carlo simulation meet.

Bayesian inverse problems (BIP) appear in a wide range of scientific and engineering fields, from protein design to digital image processing. The importance of the prior´s specification in the solution of ill-posed BIP is of utmost importance.

Recently, the interest in using so-called denoising diffusion models (DDM) as priors for solving BIP has increased rapidly.

DDM constitute a class of state-of-the-art deep generative models learning a complex data distribution by diffusing the same into a Gaussian distribution and then learning to reverse this noising process to generate synthetic data.

However, since the probability density of a DDM is intractable, the highly expressive models formed by DDM-prior-based BIP are obtained at the cost of making sampling from the posterior very challenging.

Existing approximative approaches exist, but are usually computationally intensive, suffer from non-negligible/non-controllable bias, and bear a mark of ad hoc design and lack of accuracy analysis. Thus, the present project, BIGMC, aims to develop novel Monte Carlo-guided approaches to such posterior sampling.

As a parallel goal, it aims to furnish the methodology developed with rigorous theoretical results describing its accuracy and consistency.

In view of the large potential of DDM-prior-based BIP and the widespread interest in these models, it is expected that the impact of such computationally effective and theoretically sound posterior sampling technology will be significant.