1. A computer-implemented method comprising: carrying out two or more eye-tracking processes that each provide eye-position data, wherein the eye-position data from each eye-tracking process indicates positioning of the eye over time, wherein the two or more eye-tracking processes comprise at least an ambient-glint eye-tracking process and a second eye-tracking process, wherein the ambient-glint eye-tracking process determines eye-position data based on one or more reflections of light from the eye that correspond to one or more light sources at one or more unknown locations, and wherein the eye tracking based on ambient glints determines the eye-position data using an expectation maximization process that alternatingly estimates a head-movement component and an orbital component of an observed eye movement until a combination of the estimated head-movement component and the estimated orbital component differ from an observed movement of the eye in space by less than a threshold amount; determining a reliability indication for at least one of the eye-tracking processes; determining a respective weighting value for each of two or more of the eye-tracking processes based at least in part on the reliability indication; determining a combined eye position based on a weighted combination of respective eye-position data that is concurrently available from the two or more eye-tracking processes, wherein the eye-position data from each eye-tracking process is weighted by the respectively determined weighting value for the eye-tracking process; and instructing a computing device to utilize the combined eye position for one or more functions that incorporate eye-position data.