Bayesian analysis of count-valued, binary-valued, and continuous-valued responses using unknown transformations
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Consider the situation where an analyst has a Bayesian statistical model that performs well for continuous data. However, suppose the observed data set consists of multiple response types (e.g., continuous, count-valued, Bernoulli trials, etc.), which are distributed from more than one class of distributions. We refer to these types of data as "multi-response" data sets. The goal of this article is to introduce a reasonable easy-to-implement all-purpose method that "converts" a Bayesian statistical model for continuous responses (call this the preferred model) into a Bayesian model for multi-response data sets. To do this, we consider a transformation of the data, such that the transformed data can be be reasonably modeled using the preferred model. What is unique with our strategy is that we treat the transformations as unknown and use a Bayesian approach to model this uncertainty. The implementation of our Bayesian approach to unknown transformations is straightforward, and involves two steps. The first step produces posterior replicates of the transformed data from a latent conjugate multivariate (LCM) model. The second step involves generating values from the posterior distribution implied by the preferred model. We demonstrate the flexibility of our model through an application to Bayesian additive regression trees (BART), spatial mixed effects (SME) models, and the multivariate spatio-temporal mixed effects model (MSTM). To further illustrate the potential wide use of this approach, we provide an analysis of zero-inflated records of public costs due to natural disasters obtained from the National Oceanic Atmospheric Association's (NOAA) National Centers for Environmental Information (NCEI).
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