Counting probability distributions: differential geometry and model selection

A central problem in science is deciding among competing explanations of data containing random errors. We argue that assessing the "complexity" of explanations is essential to a theoretically well-founded model selection procedure. We formulate model complexity in terms of the geometry of the space of probability distributions. Geometric complexity provides a clear intuitive understanding of several extant notions of model complexity. This approach allows us to reconceptualize the model selection problem as one of counting explanations that lie close to the "truth." We demonstrate the usefulness of the approach by applying it to the recovery of models in psychophysics.