Abstract

Statistical manifolds which were developed to provide a set of probability distributions with a differentiable structure (Amari, 1985), play a major role in the geometric study of information. These manifolds are now regarded as the  foundation for the  study of the geometry of information transfer, data analysis and quantification. These manifolds with quasi-complex structures  are  naturally appear in many geometric contexts, notably in information geometry and statistical inference. The using these methodologies is intended to facilited the quantification and extraction the best information from a statistical model (Amari and Nagaoka, 2000). Thus, the study of statistical manifolds with quasi-complex structures allows us to understand the possible interactions between information geometry and complex structures. It has been shown in (Zhang and Fei, 2018) that assuming the existence of a quasi-complex structure J on a statistical manifold (M,g,\nabla), then (M,g,J,\nabla) is (para-)K?hlerian on the condition that (J,\nabla) is a Codazzi pair. Manifolds with quasi-complex structure which a metric is Norden have been  the subject of detailed inverstigation (Leila et al., 2022). In this work, the condition for the compatibility  of the Hermitian structure and that of statistical manifolds are examined.For document, after the first section reserved for the introduction,  section 2 devoted to the presentation of the different structures is presented. Section 3 presents the main results obtained and the construction of a more general example. Finally, section 4 examines the special case of parametric statistical models of even dimension. We have also considered such a mo

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