Counting probability distributions: Differential geometry and model selection

A central problem in science is deciding among competing explanations of da ta containing random errors. We argue that assessing the "complexity" of ex planations is essential to a theoretically well-founded model selection pro cedure. We formulate model complexity in terms of the geometry of the space of probability distributions, Geometric complexity provides a clear intuit ive understanding of several extant notions of model complexity. This appro ach allows us to reconceptualize the model selection problem as one of coun ting explanations that lie close to the "truth." We demonstrate the usefuln ess of the approach by applying it to the recovery of models in psychophysi cs.