Volumes of logistic regression models with applications to model selection

The Fisher information metric is the unique Riemannian metric for statistical models which is invariant under certain natural transformations, so the corresponding geometry is likely to be important and useful for studying these models. In this paper, we focus on logistic regression models and the simplest geometric invariant, the volume. We show that the volume of a logistic regression model with $n$ observations and $q$ linearly-independent covariates is bounded below by $\pi^q$ and above by ${n \choose q} \pi^q$ when $q \le n$. We prove this with a novel generalization of the classical theorems of Pythagoras and de Gua, which is of independent interest. The finding that the volume is always finite is new, and it implies that the volume can be interpreted directly as a measure of model complexity in the minimum description length (MDL) approach to model selection. We derive an approximation to the volume and apply the resulting model-selection criterion to simulated data, giving promising results. We also show that the volume is a continuous function of the design matrix $X$ at generic $X$ but that it is discontinuous in general. This means that logistic regression models with sparse design matrices can be significantly less complex than nearby models, so our model-selection criterion favours sparse models. Lastly, for generic $X$, we show that the reparameterisation map between the natural and expectation parameter spaces induces a topological duality between certain natural polygonal decompositions of the ideal boundaries of these two spaces.