Journal of Statistical Computation and Simulation, 2015 Vol. 85, No. 1, 117–130, http://dx.doi.org/10.1080/00949655.2013.806509 Study of incompatibility or near compatibility of bivariate discrete conditional probability distributions through divergence measures Indranil Ghosh a * and N. Balakrishnan b,c a Department of Mathematics and Statistics, Austin Peay State University, Clarksville, TN, USA; b Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada; c Department of Statistics, King Abdulaziz University, Jeddah, Saudi Arabia (Received 21 September 2012; final version received 14 May 2013) 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. Keywords: incompatible conditionals; divergence measures; iterative algorithm; conditional specifica- tion; near compatibility; linear programming; non-linear programming; ε-compatibility 1. Introduction In the finite discrete case, many conditions for compatibility of conditional distributions can be derived. Such conditions provide different ways of measuring discrepancy between incompati- ble conditional distributions. The concept of ε-compatibility arises naturally in the discussion of incompatible or most nearly compatible distributions. Several approaches exist in the literature for determining the possible compatibility of two families of conditional distributions; see, for exam- ple, Arnold and Press [1] and Arnold and Gokhale.[2] Moreover, the determination of most nearly compatible or ε-compatible, as termed by Arnold and Gokhale,[2,3] has also been addressed. Here, we consider the issues of compatibility and near compatibility of given families of con- ditional distributions in the finite discrete case. In this case, a variety of compatibility conditions exist; see Arnold et al.[4,5] Based on those conditions, several different ways of measuring dis- crepancy between incompatible conditional distributions have been discussed by these authors. In addition, they have made some suggestions for defining nearly compatible distributions in incompatible cases. In this paper, we focus on the measurement of incompatibility in situations when we are given two families of conditional distributions under the discrete setup which are not compatible. We *Corresponding author. Email: ghoshi@apsu.edu © 2013 Taylor & Francis