Fisher Information and Exponential Families Parametrized by a Segment of Means

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.