1-Conformal geometry of quasi statistical manifolds

Abstract

A quasi statistical manifold is a generalization of a statistical manifold. The notion of quasi statistical manifolds was introduced to formulate the geometry of non-conservative estimating functions in statistics. Later, it was showed that quasi statistical manifolds are induced from affine distributions in the same way as statistical manifolds are induced from affine immersions. Here, an affine distribution is a non-integrable version of an affine immersion, and it is useful in quantum information geometry. On the other hand, it is known that generalized conformal geometry is useful for the study of statistical manifolds from the viewpoint of affine differential geometry. In particular, 1-conformal geometry of statistical manifolds gives a relation with the notion of affine immersions. Although generalized conformal geometry of quasi statistical manifolds is also expected to be useful, the geometry has not been cleared yet. The aim of this paper is to formulate 1-conformal geometry of quasi statistical manifolds. We research a relation between 1-conformal geometry of quasi statistical manifolds and the notion of affine distributions. As the main result, we show the fundamental theorems for affine distributions. We also formulate a hypersurface theory of quasi statistical manifolds.