The central limit theorem characterizes the limiting distribution of the sum of increasingly many finite-variance independent random variables: the limit is Gaussian. The generalized central limit theorem relaxes the finite variance assumption: the limit is Levy $\alpha$-stable. The Bayesian central limit theorem characterizes the limiting posterior under an increasingly many conditionally independent observations, assuming the posterior under each of those observations alone is everywhere twice differentiable: the limit is Gaussian. How can the twice differentiability assumption be relaxed, and what are the generalized posteriors?
Intuitively I see the Bayesian CLT as a Fourier-domain analog of the usual CLT, so I would expect the limiting posterior to be the Fourier transform of a stable distribution. Examples likelihoods are normal (scaling as $\alpha=2$), Laplace (scaling as $\alpha=1$), and Uniform(0,1) (scaling as $\alpha=\infty$).
