Fisher Information and Exponential Families Parametrized by a Segment of Means

Abstract

We consider natural and general exponential families $(Q_m{)}_{m\in M}$ on ${\mathbb{R}}^d$ parametrized by the means. We study the submodels $(Q_{\theta m_1+(1-\theta )m_2}{)}_{\theta \in [0,1]}$ parametrized by a segment in the means domain, mainly from the point of view of the Fisher information. Such a parametrization allows for a parsimonious model and is particularly useful in practical situations when hesitating between two parameters $m_1$ and $m_2$. The most interesting examples are obtained when ${\mathbb{R}}^d$ is a linear space of matrices, in particular for Gaussian and Wishart models.