We study the stability of posterior predictive inferences to the specification of the likelihood model and perturbations of the data generating process. In modern big data analyses, the decision-maker may elicit useful broad structural judgements but a level of interpolation is required to arrive at a likelihood model. One model, often a computationally convenient canonical form, is chosen, when many alternatives would have been equally consistent with the elicited judgements. Equally, observational datasets often contain unforeseen heterogeneities and recording errors. Acknowledging such imprecisions, a faithful Bayesian analysis should be stable across reasonable equivalence classes for these inputs. We show that traditional Bayesian updating provides stability across a very strict class of likelihood models and DGPs, while a generalised Bayesian alternative using the beta-divergence loss function is shown to be stable across practical and interpretable neighbourhoods. We illustrate this in linear regression, binary classification, and mixture modelling examples, showing that stable updating does not compromise the ability to learn about the DGP. These stability results provide a compelling justification for using generalised Bayes to facilitate inference under simplified canonical models.

翻译：我们研究了后验预测推断对似然模型设定和数据生成过程扰动的稳定性。在现代大数据分析中，决策者可以提炼出有用的宽泛结构判断，但需要一定程度的插值来构建似然模型。当多个备选模型与提炼出的判断同样一致时，通常只选择一个计算上便利的规范形式模型。同样，观测数据集往往包含不可预见的异质性和记录误差。承认这些不精确性，忠实的贝叶斯分析应在这些输入的合理等价类中保持稳定。我们证明，传统贝叶斯更新仅在非常严格的似然模型和数据生成过程类别中具有稳定性，而使用β散度损失函数的广义贝叶斯替代方法在实际且可解释的邻域内保持稳定。我们在线性回归、二元分类和混合建模实例中进行了说明，表明稳定更新不会损害对数据生成过程的学习能力。这些稳定性结果为使用广义贝叶斯在简化规范模型下进行推断提供了令人信服的理由。