Conditional sampling within generative diffusion models

As an example, when the density function of π( | y) is available (up to a constant), Markov chain Monte Carlo (MCMC, Meyn and Tweedie, 2009) methods are popular and generic algorithms widely used. The MCMC algorithms simulate a Markov chain that leaves the target distribution invariant. The drawback is that this often makes the algorithms computationally and statistically inefficient for high-dimensional problems. In this article, we discuss an emerging class of samplers that leverage generative diffusions (see, e.g., Benton et al., 2024; Song et al., 2021), which have empirically worked well for many Bayesian inverse problems. At the heart, the generative diffusions aim to find a continuos-time Markov process (e.g., stochastic differential equation) that bridges the target distribution and a reference measure, so that sampling the target simplifies to sample the reference and the Markov process. In contrast to traditional samplers such as MCMC which use the target's density function to build statistically exact samplers, the generative diffusions use the data to approximate a sampler akin to normalising flow (Chen et al., 2018; Papamakarios et al., 2021) and flow matching (Lipman et al., 2023). This comes with at least three benefits compared to MCMC: 1) scalability of the problem dimension (after the training time), 2) no need to explicitly know the target density function, 3) and the resulting samplers are embarrassingly differentiable (see a use case in Watson et al., 2022). However, the generative diffusion framework (for unconditional sampling) is not immediately applicable to conditional sampling, since we do not have the conditional data samples from π( | y) required to train the generative samplers.