One of the major problems in modeling natural signals is that signals with very similar structure may locally have completely different measurements, e.g., images taken under different illumination conditions, or the speech signal captured in different environments. While there have been many successful attempts to address these problems in application-specific settings, we believe that underlying a large set of problems in signal representation is a representational deficiency of intensity-derived local measurements that are the basis of most efficient models. We argue that interesting structure in signals is better captured when the signal is defined as a matrix whose entries are discrete indices to a separate palette of possible measurements. In order to model the variability in signal structure, we define a signal class not by a single index map, but by a probability distribution over the index maps, which can be estimated from the data, and which we call probabilistic index maps. The existing algorithms can be adapted to work with this representation. Furthermore, the probabilistic index map representation leads to algorithms with computational costs proportional to either the size of the palette or the log of the size of the palette, making the cost of significantly increased invariance to non-structural changes quite bearable. We illustrate the benefits of the probabilistic index map representation in several applications in computer vision and speech processing.