A concurrence theorem for alpha-connections on the space of t-distributions and its application

Abstract

The space of Student's t-distributions with $\nu$ degrees of freedom is the upper half-plane $\mathbb H$ with the location-scale coordinates. A normal distribution is a t-distribution with $\nu=\infty$. The $\alpha$-connections for t-distributions form a line ${\mathbb L}^\nu$ in the space of affine connections on $\mathbb H$. We show that the family $\{{\mathbb L}^\nu\}_{\nu\in(1,\infty]}$ has the concurrent point which presents the e-connection for normal distributions. As an application, generalizing the previous result of the author, we construct a contact Hamiltonian flow which visualizes certain Bayesian learnings.