In this thesis, we investigate mappings from primary sample space to path space and their inverses. We give special consideration to how these mappings interact with Markov Chain Monte Carlo rendering methods operating in primary sample space. In particular, we show that such methods perform uncontrolled changes to light transport paths in the presence of geometric and material discontinuities, discrete choices and multiple sampling techniques. We also show that ignoring internals of these mappings can lead to poor noise distribution on the image plane and low acceptance rates for large steps. Our contributions are three-fold: We describe how to construct inverses of several path sampling techniques employed in graphics in order to robustly turn transport paths back into the random numbers that produced them, and use these inverses to create two new perturbations for Multiplexed Metropolis Light Transport; we introduce Multiple Correlated-Try Metropolis to Markov Chain Monte Carlo rendering and apply it to create a new large step mutation based on bidirectional connections; and we show how annotating dimensions of primary sample space with information about how they are employed in the path sampling process can make Markov Chain Monte Carlo rendering methods more robust, and give two applications of such an approach.