Contraction rates for generalized posteriors based on $f$-divergences: a diffusion process approach

We study the finite-sample behavior of generalized posteriors defined via $f$-divergences, a broad class of posteriors that includes the standard Bayesian posterior along with most of its generalizations. Our main contribution is to extend the Langevin diffusion representation of the Bayesian posterior to this broader class. With this perspective, we obtain non-asymptotic posterior contraction rates for $f$-divergence-based posteriors by bounding the moments of their associated diffusion. Our results establish nearly optimal rates and clarify how different divergence choices influence posterior concentration. Finally, we illustrate the general framework with concrete examples.