Study of incompatibility or near compatibility of bivariate discrete conditional probability distributions through divergence measures

Consider a two-dimensional discrete random variable (X, Y) with possible values 1, 2, …, I for X and 1, 2, …, J for Y. For specifying the distribution of (X, Y), suppose both conditional distributions, of X given Y and of Y given X, are provided. Under this setting, we present here different ways of measuring discrepancy between incompatible conditional distributions in the finite discrete case. In the process, we also suggest different ways of defining the most nearly compatible distributions in incompatible cases. Many new divergence measures are discussed along with those that are already known for determining the most nearly compatible joint distribution P. Finally, a comparative study is carried out between all these divergence measures as some examples.