The Complexity of the Dirichlet Model for Multiple Alignment Data

A model is a set of possible theories for describing a set of data. When the data are used to select a maximum-likelihood theory, an important question is how many effectively independent theories the model contains; the log of this number is called the model's complexity. The Dirichlet model is the set of all Dirichlet distributions, which are probability densities over the space of multinomials. A Dirichlet distribution may be used to describe multiple-alignment data, consisting of n columns of letters, with c letters in each column. We here derive, in the limit of large n and c, a closed-form expression for the complexity of the Dirichlet model applied to such data. For small c, we derive as well a minor correction to this formula, which is easily calculated by Monte Carlo simulation. Although our results are confined to the Dirichlet model, they may cast light as well on the complexity of Dirichlet mixture models, which have been applied fruitfully to the study of protein multiple sequence alignments.