Current searches for new physics are often guided by beyond the Standard Model (BSM) theories that depend on many unknown parameters, which makes testing predictions computationally challenging. Bayesian neural networks (BNNs) can map the parameter space of these theories to a meaningful distribution of observables. The use of these networks is demonstrated by the new package TensorBNN by modeling the predictions of the phenomenological Minimal Supersymmetric Standard Model (pMSSM), a BSM theory with 19 free parameters. The predicted quantities are cross sections for arbitrary pMSSM parameter points, the mass of the associated lightest neutral Higgs boson, and the theoretical viability of the parameter points. All three quantities are modeled with average percent errors of 3.3% or less in a time orders of magnitude shorter than the supersymmetry codes from which the results are derived. The posterior densities, provided as point clouds, provide meaningful Bayesian confidence intervals for the predictions, further demonstrating the potential for machine learning to accurately model the mapping from the high dimensional spaces of BSM theories to their predictions.