🤖 AI Summary
This work rigorously establishes that predictive Bayesian inference (PBI) suffers from severely miscalibrated posterior uncertainty in practice when the forward predictive model is misspecified, potentially yielding credible sets with coverage approaching zero. We prove for the first time that the posterior concentrates precisely on the target dictated solely by the chosen forward predictive model, and that calibration of inference is guaranteed only when this model fully encompasses the true data-generating mechanism. By integrating predictive recursion algorithms, Bayesian nonparametric theory, and posterior concentration analysis, we demonstrate the fundamental role of the predictive model in determining inferential reliability and delineate the necessary conditions under which PBI can deliver valid uncertainty quantification.
📝 Abstract
Predictive Bayesian inference (PBI) represents a model-and prior-agnostic approach to standard Bayesian inference which allows users to quantify uncertainty for a functional of interest only by specifying a forward predictive model for future unobserved data. The flexibility and generality of this framework have led to a host of novel algorithms for implementing this approach, and many empirical applications, yet the reliability of the resulting inferences for the underlying statistical functional of interest remains unclear. Herein, we demonstrate that when using PBI for a population functional of interest, the resulting posterior concentrates onto a well-defined quantity that explicitly depends on the forward predictive model used to implement the predictive recursion underlying the method. Furthermore, the forward predictive model entirely determines the uncertainty quantification produced in PBI. Consequently, our results show that if the predictive model does not capture all relevant features of the data, and, even in very simple examples, the coverage of predictive Bayes credible sets for the population value of the functional of interest can be arbitrarily close to zero. We carefully explain why this occurs, and show that this behavior is directly tied to the inaccuracy of the forward predictive model used to produce future observations within the PBI framework. As a consequence, our results imply that in order for PBI to deliver calibrated posterior inferences, the resulting predictive engine used to generate posterior samples must contain, in a well-defined sense, the true DGP, else inferences generated under this framework will not be calibrated.