Remote testing platforms have proven to be a viable tool for hearing research, but the ability to obtain efficient and reliable data remains a significant barrier. Adaptive up-down procedures are widely used to estimate hearing abilities because they can be performed easily with minimal assumptions about underlying mechanisms. However, they are inefficient compared to parametric alternatives, and the optimization of starting level, step size, stopping rule, and other factors continues to rely on trial and error. This reflects the limitations of our theoretical understanding, which cannot explain the complex, multi-component error patterns observed in thresholds obtained in controlled simulations. The present study reports on the development of a mathematical framework for nonparametric psychophysical procedures. A method will be described for creating models of the combined listener-procedure system as an equivalence class of directed graphs. The model can be used to calculate the trial-by-trial and asymptotic stimulus distributions for ascending and descending trials, the rate of convergence, and other fundamental properties. Using this approach, stimulus selection, stopping, and threshold calculation can be optimized with typical minimization procedures. Insights about the mathematical structure of psychophysical procedures promise to drive the development of novel tools to address the needs of remote test administration.