Inference functions in high dimensional Bayesian inference

Nonparametric Bayesian models, such as those based on the Dirichlet process or its many variants, provide a flexible class of models that allow us to fit widely varying patterns in data. Typical uses of the models include relatively lowdimensional driving terms to capture global features of the data along with a nonparametric structure to capture local features. The models are particularly good at handling outliers, a common form of local behavior, and examination of the posterior often shows that a portion of the model is chasing the outliers. This suggests the need for robust inference to discount the impact of the outliers on the overall analysis. We advocate the use of inference functions to define relevant parameters that are robust to the deficiencies in the model and illustrate their use in two examples.