The predictive nature of Bayesian inference

We explore the predictive viewpoint of Bayesian inference that bypasses the conventional reliance on priors and likelihoods. By treating the joint predictive distribution of observables as the fundamental element, we gain fresh insights into the nature of Bayesian reasoning and derive principled generalizations – including new methods such as martingale posteriors. The predictive perspective improves our understanding of uncertainty quantification and can facilitate more adaptable, data-driven approaches to probabilistic modelling.