On the Convergence of Iterative Algorithms Involving Incompatible/Minimally Compatible Finite Discrete Conditionals: Short Proof

What the paper says

To search for the existence of most nearly compatible probability distributions under the discrete set-up (when the given putative conditional distributions do not yield a joint distribution), there are ample evidences that exist in the literature which suggests to use a wide collection of divergence measures as pseudo-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] have utilized several of such divergence measures, such as the Power divergence, $$\chi^{2}$$ -divergence, modified Renyi’s divergence among several other measures in this regard. Noticeably, the procedure in search for a most nearly compatible probability matrix, P, involves an iterative algorithm based on those divergence measures. Although, in all numerical evaluations with varying choices of the model parameters (as appropriate) have been found to be convergent, see, in the works of [9, 11, 12] among others, a formal mathematical proof on the issue of convergence is still lacking. Recently, some progress has been made in this direction, when the number of rows is equal to the number of columns for the given two conditionals, see, [10]. Since iterative algorithm for different divergence measures do not follow a single mechanism, the corresponding mathematical proof(s) will be different. Motivated by this rationale, in this article, we put forward a sketch of the proof regarding the convergence of the iterative algorithm(s) for the power divergence measure and another divergence measure, both of which involve a model parameter $$\lambda.$$ The proof for the more general case still remains an open problem.