High-resolution probabilistic projections of precipitation and temperature under climate change are crucial for stakeholders to make well-informed decisions in mitigating and adapting to more intense, longer duration, and more frequent extreme weather events. General circulation models (GCMs) provide us with the data to study climate change at the continental spatial scales, but are too coarse for local adaption. Furthermore, ensembles of multiple models, initial conditions, and emission trajectories must be harnessed for well quantified probabilistic estimates. Statistical downscaling, an approach that learns a functional mapping between low- and high-resolution GCMs, can be used to generate high-resolution ensemble projections in a computationally efficient manner. However, this process exacerbates, at a local scale, uncertainties inherently found in GCMs. Hence, it is crucial for our statistical downscaling methods to incorporate and quantify uncertainties, including both epistemic, or parameter misunderstanding, and aleatoric, or observational, uncertainties. In this work, we present a Bayesian deep learning and image super-resolution approach for statistical downscaling using discrete-continuous and non-normal likelihoods. Promising results for downscaling daily precipitation in the contiguous United States measured on predictive accuracy and uncertainty quantification are presented. Future work on stacking Bayesian deep learning networks and harnessing ensembles of high-resolution GCMs is discussed.