# On the role of the overall effect in exponential families

## TL;DR

This paper examines how adding or removing the overall effect in exponential families affects their properties, geometry, and computational aspects, with implications for statistical modeling and biological data analysis.

## Contribution

It characterizes the impact of the overall effect on exponential family properties, geometry, and algorithms, linking algebraic geometry concepts to statistical modeling.

## Key findings

- Adding the overall effect creates the smallest regular exponential family containing the curved one.
- Removing the overall effect simplifies the family but can lead to different estimation properties.
- Including the overall effect can produce estimates outside the intended model in biological applications.

## Abstract

Exponential families of discrete probability distributions when the normalizing constant (or overall effect) is added or removed are compared in this paper. The latter setup, in which the exponential family is curved, is particularly relevant when the sample space is an incomplete Cartesian product or when it is very large, so that the computational burden is significant. The lack or presence of the overall effect has a fundamental impact on the properties of the exponential family. When the overall effect is added, the family becomes the smallest regular exponential family containing the curved one. The procedure is related to the homogenization of an inhomogeneous variety discussed in algebraic geometry, of which a statistical interpretation is given as an augmentation of the sample space. The changes in the kernel basis representation when the overall effect is included or removed are derived. The geometry of maximum likelihood estimates, also allowing zero observed frequencies, is described with and without the overall effect, and various algorithms are compared. The importance of the results is illustrated by an example from cell biology, showing that routinely including the overall effect leads to estimates which are not in the model intended by the researchers.