  We study methods for reconstructing Bayesian uncertainties on dynamical mass
estimates of galaxy clusters using convolutional neural networks (CNNs). We
discuss the statistical background of approximate Bayesian neural networks and
demonstrate how variational inference techniques can be used to perform
computationally tractable posterior estimation for a variety of deep neural
architectures. We explore how various model designs and statistical assumptions
impact prediction accuracy and uncertainty reconstruction in the context of
cluster mass estimation. We measure the quality of our model posterior recovery
using a mock cluster observation catalog derived from the MultiDark simulation
and UniverseMachine catalog. We show that approximate Bayesian CNNs produce
highly accurate dynamical cluster mass posteriors. These model posteriors are
log-normal in cluster mass and recover $68\%$ and $90\%$ confidence intervals
to within $1\%$ of their measured value. We note how this rigorous modeling of
dynamical mass posteriors is necessary for using cluster abundance measurements
to constrain cosmological parameters.
