Diffusion models are all the rage in machine learning these days. Today Laurence Levasseur (Montréal) gave a beautiful talk at the CZS Summer School about how diffusion works. She started with a long physics introduction, which was great, and also insightful, about how diffusion works in small physical systems. Then she showed how it can be turned into a method for sampling very difficult probability distributions.
I have a history of working on MCMC methods. These permit you to sample a posterior pdf when you only know a function f that is related to your posterior pdf by some unknown normalization constant. Similarly, diffusion lets you sample from a pdf when you only know the gradient of f. Again, you don't need the normalization. That makes me wonder: Should we be using diffusion in places where we currently use MCMC? I bet the answer is yes, for at least some problems.
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