We consider the problem of robust Bayesian inference on the mean regression function allowing the residual density to change flexibly with predictors. The proposed class of models is based on a Gaussian process (GP) prior for the mean regression function and mixtures of Gaussians for the collection of residual densities indexed by predictors. Initially considering the homoscedastic case, we propose priors for the residual density based on probit stick-breaking mixtures. We provide sufficient conditions to ensure strong posterior consistency in estimating the regression function, generalizing existing theory focused on parametric residual distributions. The homoscedastic priors are generalized to allow residual densities to change nonparametrically with predictors through incorporating GP in the stick-breaking components. This leads to a robust Bayesian regression procedure that automatically down-weights outliers and influential observations in a locally adaptive manner. The methods are illustrated using simulated and real data applications.

中文翻译：

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具有不同残差密度的贝叶斯非参数回归

我们考虑了对平均回归函数的鲁棒贝叶斯推理问题，允许残差密度随预测变量灵活变化。建议的模型类别基于高斯过程 (GP) 先验的平均回归函数和高斯混合，用于收集由预测变量索引的残余密度。最初考虑同方差情况，我们提出了基于概率棒断裂混合物的残余密度的先验。我们提供了足够的条件来确保在估计回归函数时具有很强的后验一致性，概括了现有的专注于参数残差分布的理论。同方差先验被概括为允许残余密度通过在断棒组件中结合 GP 随预测变量非参数地改变。这导致了稳健的贝叶斯回归过程，该过程以局部自适应方式自动降低异常值和有影响的观察值的权重。使用模拟和真实数据应用程序来说明这些方法。