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.

翻译：预测贝叶斯推断（PBI）是一种与模型和先验无关的标准贝叶斯推断方法，用户仅需通过指定未来未观测数据的正向预测模型，即可量化目标泛函的不确定性。该框架的灵活性与通用性催生了众多实现该算法的新方法及大量实证应用，但其对底层统计目标泛函的推断可靠性仍不明确。本文证明，当对群体目标泛函使用PBI时，其后验分布会聚集于一个明确界定的量，该量显式依赖于实现方法中预测递归过程的正向预测模型。此外，正向预测模型完全决定了PBI中产生的不确定性量化结果。因此，我们的结果表明：若预测模型未能捕捉数据的所有相关特征，即使在极简示例中，针对该泛函群体值的预测贝叶斯可信集的覆盖概率也可能趋近于零。我们详细解释了这一现象的成因，并证明该行为直接源于PBI框架中用于生成未来观测的正向预测模型的不准确性。由此得出推论：为使PBI提供校准的后验推断，用于生成后验样本的预测引擎必须在严格意义上包含真实数据生成过程（DGP），否则该框架下产生的推断将无法实现校准。