Here, we present the application of an advanced sparse Gaussian process-based machine learning algorithm to the challenge of predicting the yields of inertial confinement fusion (ICF) experiments. The algorithm is used to investigate the parameter space of an extremely robust ICF design for the National Ignition Facility, the ``Simplest Design''; deuterium-tritium gas in a plastic ablator with a Gaussian, Planckian drive. In particular, we show that: 1) GPz has the potential to decompose uncertainty on predictions into uncertainty from lack of data and shot-to-shot variation; 2) it permits the incorporation of science-goal-specific cost-sensitive learning (CSL), e.g., focusing on the high-yield parts of parameter space; and 3) it is very fast and effective in high dimensions.