In the usual Bayesian setting, a full probabilistic model is required to link the data and parameters, and the form of this model and the inference and prediction mechanisms are specified via de Finetti's representation. In general, such a formulation is not robust to model misspecification of its component parts. An alternative approach is to draw inference based on loss functions, where the quantity of interest is defined as a minimizer of some expected loss, and to construct posterior distributions based on the loss-based formulation; this strategy underpins the construction of the Gibbs posterior. We develop a Bayesian non-parametric approach; specifically, we generalize the Bayesian bootstrap, and specify a Dirichlet process model for the distribution of the observables. We implement this using direct prior-to-posterior calculations, but also using predictive sampling. We also study the assessment of posterior validity for non-standard Bayesian calculations. We show that the developed non-standard Bayesian updating procedures yield valid posterior distributions in terms of consistency and asymptotic normality under model misspecification. Simulation studies show that the proposed methods can recover the true value of the parameter under misspecification.

翻译：在传统贝叶斯框架中，需要建立完整的概率模型来连接数据与参数，该模型的形式及其推断与预测机制均通过德菲内蒂表示定理进行设定。一般而言，此类构建方式对其各组成部分的模型误设缺乏稳健性。另一种替代方案是基于损失函数进行推断：将目标量定义为某种期望损失的最小化解，并基于损失函数框架构建后验分布；该策略构成了吉布斯后验构建的理论基础。本文提出一种非参数贝叶斯方法：具体而言，我们推广了贝叶斯自助法，并为观测变量的分布设定了狄利克雷过程模型。我们通过直接先验-后验计算与预测抽样两种方式实现该模型。同时研究了非标准贝叶斯计算的后验有效性评估问题。我们证明：在模型误设条件下，所发展的非标准贝叶斯更新程序能产生具有一致性及渐近正态性的有效后验分布。仿真研究表明，所提方法在模型误设情况下仍能恢复参数的真实值。