Volumes of logistic regression models with applications to model selection

TL;DR

This paper investigates the geometric properties of logistic regression models, establishing bounds on their Fisher information volumes, which serve as a measure of model complexity, and explores implications for model selection and sparsity.

Contribution

It introduces bounds on Fisher information volumes for logistic regression, generalizes classical theorems, and reveals how volume discontinuities influence model sparsity preferences.

Findings

Fisher information volume is bounded between π^q and (n choose q)π^q.

Volume is a continuous function of the design matrix at generic points.

Models with sparse design matrices can be significantly less complex, favoring sparsity.

Abstract

Logistic regression models with $n$ observations and $q$ linearly-independent covariates are shown to have Fisher information volumes which are bounded below by $\pi^q$ and above by ${n \choose q} \pi^q$. This is proved 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 directly interpreted as a measure of model complexity. The volume is shown to be a continuous function of the design matrix $X$ at generic $X$, but to be discontinuous in general. This means that models with sparse design matrices can be significantly less complex than nearby models, so the resulting model-selection criterion prefers sparse models. This is analogous to the way that $\ell^1$-regularisation tends to prefer sparse model fits, though in our case this behaviour arises spontaneously from general principles. Lastly, an unusual topological duality is shown to exist between the ideal boundaries of the natural and expectation parameter spaces of logistic regression models.