Abstract
To search for the existence of most nearly compatible probabilitydistributions under the discrete set-up (when the given putativeconditional distributions do not yield a joint distribution),there are ample evidences that exist in the literature whichsuggests to use a wide collection of divergence measures aspseudo-distance (equivalently as measures of dissimilarity)measures. A well-documented reference can be found in see [1, 13,14] and the references cited therein. Independently, [11, 12] haveutilized several of such divergence measures, such as the Powerdivergence, \(\chi^{2}\) -divergence, modified Renyi’s divergenceamong several other measures in this regard. Noticeably, theprocedure in search for a most nearly compatible probabilitymatrix, P, involves an iterative algorithm based on thosedivergence measures. Although, in all numerical evaluations withvarying choices of the model parameters (as appropriate) have beenfound to be convergent, see, in the works of [9, 11, 12] amongothers, a formal mathematical proof on the issue of convergence isstill lacking. Recently, some progress has been made in thisdirection, when the number of rows is equal to the number ofcolumns for the given two conditionals, see, [10]. Since iterativealgorithm for different divergence measures do not follow a singlemechanism, the corresponding mathematical proof(s) will bedifferent. Motivated by this rationale, in this article, we putforward a sketch of the proof regarding the convergence of theiterative algorithm(s) for the power divergence measure andanother divergence measure, both of which involve a modelparameter \(\lambda.\) The proof for the more general case stillremains an open problem.
