This dissertation contributes to Bayesian statistics and economics using latent variable methods. The first chapter explores interweaving methods for constructing Markov chains in dynamic linear models (DLMs). Here, several new data augmentations are defined for the DLM, and a negative result concerning the sort of augmentations that can be found for the model is proved. A simulation study using a specific DLM illuminates when each of several DA and interweaving algorithms performs well. The second chapter is an extention of the first, introducing a method to extend the results of the first chapter to DLMs where the observation level matrix is not square. Finally, the last chapter develops methods for Bayesian causal inference to compare two treatments using partial identification methods. Specifically, it develops priors that capture the intuition of standard partial identification methods in the Bayesian setting and extends those prior to a hierarchical setting. Then it illustrates how to use the model with these priors in an example evaluating the effectiveness of the National School Lunch Program.