  Models for heteroskedastic data are relevant in a wide variety of
applications ranging from financial time series to environmental statistics.
However, the topic of modeling the variance function conditionally has not seen
near as much attention as modeling the mean. Volatility models have been used
in specific applications, but these models can be difficult to fit in a
Bayesian setting due to posterior distributions that are challenging to sample
from efficiently. In this work, we introduce a general model for
heteroskedastic data. This approach models the conditional variance in a mixed
model approach as a function of any desired covariates or random effects. We
rely on new distribution theory in order to construct priors that yield fully
conjugate full conditional distributions. Thus, our approach can easily be fit
via Gibbs sampling. Furthermore, we extend the model to a deep learning
approach that can provide highly accurate estimates for time dependent data. We
also provide an extension for heavy-tailed data. We illustrate our methodology
via three applications. The first application utilizes a high dimensional soil
dataset with inherent spatial dependence. The second application involves
modeling of asset volatility. The third application focuses on clinical trial
data for creatinine.
