On warped product on information geometry: statistical manifolds and statistical models

We study a differential geometric construction, the warped product, on the background geometry for information theory. Divergences, dual structures and symmetric 3-tensor are studied under this construction, and we show that warped product of manifolds endow with such structure also is endowed with the same geometric notion. However, warped product does not preserve canonical divergences, which in particular shows that warped product lacks of meaning in the information theory setting.