A nice result of Doob from the 1940s showed how Bayesian inference can be understood by predictive sampling. In this framework, which does not necessarily need to start with a prior, martingales become the key tool for ensuring convergence of limits of variables which can be treated as samples from the posterior distribution. The practical features of the martingales are that very complex models requiring MCMC, for example, can be sampled directly and can moreover make use of parallel sampling. Some Bayesian nonparametric problems will be illustrated.

The paper has been accepted with discussion by the Journal of the Royal Statistical Society, Series B

You can find the final version of the paper here: https://arxiv.org/abs/2103.15671