Bounding the best achievable error probability for binary classification problems is relevant to many applications including machine learning, signal processing, and information theory. Many bounds on the Bayes binary classification error rate depend on information divergences between the pair of class distributions. Recently, the Henze–Penrose (HP) divergence has been proposed for bounding classification error probability. We consider the problem of empirically estimating the HP-divergence from random samples. We derive a bound on the convergence rate for the Friedman–Rafsky (FR) estimator of the HP-divergence, which is related to a multivariate runs statistic for testing between two distributions. The FR estimator is derived from a multicolored Euclidean minimal spanning tree (MST) that spans the merged samples. We obtain a concentration inequality for the Friedman–Rafsky estimator of the Henze–Penrose divergence. We validate our results experimentally and illustrate their application to real datasets.

Predicting ncRNA–protein interactions (NPIs) is essential for understanding regulatory roles in cellular processes and disease mechanisms, yet experimental methods are costly and time-consuming. In this study, we propose NPI-WGNN, a novel weighted graph neural network model designed to enhance NPI prediction by incorporating topological insights from graph structures. Our approach introduces a bipartite version of the high-order common neighbor (HOCN) similarity metric to assign edge weights in an ncRNA–protein network, refining node embeddings via weighted node2vec. We further enrich these embeddings with centrality measures, such as degree and Katz centralities, to capture network hierarchy and connectivity. To optimize prediction accuracy, we employ a hybrid GNN architecture that combines graph convolutional network (GCN), graph attention network (GAT), and GraphSAGE layers, each contributing unique advantages: GraphSAGE offers scalability, GCN provides a global structural perspective, and GAT applies dynamic neighbor weighting. An ablation study confirms the complementary strengths of these layers, showing that their integration improves predictive accuracy and robustness across varied graph complexities. Experimental results on three benchmark datasets demonstrate that NPI-WGNN outperforms state-of-the-art methods, achieving up to 96.1% accuracy, 97.5% sensitivity, and an F1-score of 0.96, positioning it as a robust and accurate framework for ncRNA–protein interaction prediction.

The generalized belief propagation (GBP), introduced by Yedidia et al., is an extension of the belief propagation (BP) algorithm, which is widely used in different problems involved in calculating exact or approximate marginals of probability distributions. In many problems, it has been observed that the accuracy of GBP considerably outperforms that of BP. However, because in general the computational complexity of GBP is higher than BP, its application is limited in practice. In this paper, we introduce a stochastic version of GBP called stochastic generalized belief propagation (SGBP) that can be considered as an extension to the stochastic BP (SBP) algorithm introduced by Noorshams et al. They have shown that SBP reduces the complexity per iteration of BP by an order of magnitude in alphabet size. In contrast to SBP, SGBP can reduce the computation complexity if certain topological conditions are met by the region graph associated to a graphical model. However, this reduction can be larger than only one order of magnitude in alphabet size. In this paper, we characterize these conditions and the amount of computation gain that we can obtain by using SGBP. Finally, using similar proof techniques employed by Noorshams et al., for general graphical models satisfy contraction conditions, we prove the asymptotic convergence of SGBP to the unique GBP fixed point, as well as providing non-asymptotic upper bounds on the mean square error and on the high probability error.