Discussions of the foundations of perceptual inference have often centered on 2 governing principles, the likelihood principle and the simplicity principle. Historically, these principles have usually been seen as opposed, but contemporary statistical (e.g., Bayesian) theory tends to see them as consistent, because for a variety of reasons simpler models (i.e., those with fewer dimensions or free parameters) make better predictors than more complex ones. In perception, many interpretation spaces are naturally hierarchical, meaning that they consist of a set of mutually embedded model classes of various levels of complexity, including simpler (lower dimensional) classes that are special cases of more complex ones. This article shows how such spaces can be regarded as algebraic structures, for example, as partial orders or lattices, with interpretations ordered in terms of dimensionality. The natural inference rule in such a space is a kind of simplicity rule: Among all interpretations qualitatively consistent with the image, draw the one that is lowest in the partial order, called the maximum-depth interpretation. This interpretation also maximizes the Bayesian posterior under certain simplifying assumptions, consistent with a unification of simplicity and likelihood principles. Moreover, the algebraic approach brings out the compositional structure inherent in such spaces, showing how perceptual interpretations are composed from a lexicon of primitive perceptual descriptors. (Contains 3 footnotes and 6 figures.)