We advocate for a new statistical principle that combines the most desirable aspects of both parameter inference and density estimation. This leads us to the predictively oriented (PrO) posterior, which expresses uncertainty as a consequence of predictive ability. Doing so leads to inferences which predictively dominate both classical and generalised Bayes posterior predictive distributions: up to logarithmic factors, PrO posteriors converge to the predictively optimal model average at rate $n^{-1/2}$. Whereas classical and generalised Bayes posteriors only achieve this rate if the model can recover the data-generating process, PrO posteriors adapt to the level of model misspecification. This means that they concentrate around the true model at rate $n^{1/2}$ in the same way as Bayes and Gibbs posteriors if the model can recover the data-generating distribution, but do \textit{not} concentrate in the presence of non-trivial forms of model misspecification. Instead, they stabilise towards a predictively optimal posterior whose degree of irreducible uncertainty admits an interpretation as the degree of model misspecification -- a sharp contrast to how Bayesian uncertainty and its existing extensions behave. Lastly, we show that PrO posteriors can be sampled from by evolving particles based on mean field Langevin dynamics, and verify the practical significance of our theoretical developments on a number of numerical examples.

翻译：我们提出一种新的统计原则，该原则融合了参数推断与密度估计中最理想的特性。这引导我们提出预测导向（PrO）后验，其将不确定性表达为预测能力的自然结果。由此得到的推断在预测性能上优于经典贝叶斯与广义贝叶斯后验预测分布：在忽略对数因子的前提下，PrO后验以$n^{-1/2}$的速率收敛于预测最优的模型平均。而经典与广义贝叶斯后验仅当模型能够还原真实数据生成过程时才能达到此收敛速率，PrO后验则能自适应于模型误设的程度。这意味着若模型能够还原真实数据生成分布，PrO后验会以$n^{1/2}$的速率围绕真实模型集中，其表现与贝叶斯后验及吉布斯后验相同；但在存在非平凡模型误设的情况下，它们\emph{不会}过度集中。相反，它们会稳定收敛于一个预测最优的后验分布，其不可约不确定性的程度可解释为模型误设的程度——这与贝叶斯不确定性及其现有扩展的行为形成鲜明对比。最后，我们证明可通过基于平均场朗之万动力学的粒子演化方法对PrO后验进行采样，并通过若干数值算例验证了理论发展的实际意义。