Reverse-BSDE Monte Carlo
This work provides a novel sampling approach for Bayesian statistics, shifting from learning to sampling, but it appears incremental as it builds on existing diffusion models.
The paper tackled the problem of sampling from high-dimensional distributions with densities known up to a normalization constant, common in Bayesian statistics, by reformulating diffusion-based generative models as a Forward-Backward Stochastic Differential Equation (FBSDE) to avoid pre-estimating gradient of log target density, and proposed a numerical solution using deep learning techniques.
Recently, there has been a growing interest in generative models based on diffusions driven by the empirical robustness of these methods in generating high-dimensional photorealistic images and the possibility of using the vast existing toolbox of stochastic differential equations. %This remarkable ability may stem from their capacity to model and generate multimodal distributions. In this work, we offer a novel perspective on the approach introduced in Song et al. (2021), shifting the focus from a "learning" problem to a "sampling" problem. To achieve this, we reformulate the equations governing diffusion-based generative models as a Forward-Backward Stochastic Differential Equation (FBSDE), which avoids the well-known issue of pre-estimating the gradient of the log target density. The solution of this FBSDE is proved to be unique using non-standard techniques. Additionally, we propose a numerical solution to this problem, leveraging on Deep Learning techniques. This reformulation opens new pathways for sampling multidimensional distributions with densities known up to a normalization constant, a problem frequently encountered in Bayesian statistics.