Bayesian approaches for inference after selection and model fitting
This thesis presents a set of methods unified around the theme of providing valid inference when data are used to answer multiple questions of interest. The first portion takes on the case where the data are used twice, first to select targets of inference, and then a second time to form estimates for these targets. The proposed method uses a Bayesian formulation to give more efficient (shorter) confidence intervals which properly account for selection in order to retain nominal frequentist coverage. The second portion, comprising the bulk of this thesis, formalizes the approach of posterior summarization, unifying an set of ideas originating from the early 2000s. Posterior summarization is the process by which a model is fit to the relevant underlying outcome, and the model is interpreted through a post hoc explortation via lower-dimensional functionals. The data are used only once, to fit the model in the first stage. This approach is applied to interpret predictive trends within nonparametric regression models, select important confounders and perform model specification sensitivity analyses in linear models for causal effect estimation, and detect the presence of heterogeneous treatment effects in observational studies. These methods are applied to several real and simulated datasets.