BPFNN: Bayesian Probabilistic Fuzzy Neural Networks for Uncertainty-Aware Clustering and Probabilistic Fuzzy Reasoning

This article introduces the Bayesian probabilistic fuzzy neural network (BPFNN), a unified architecture designed to overcome the challenges of conventional fuzzy clustering and neural networks in terms of uncertainty, noise, and interpretability. At its core, the Bayesian probabilistic fuzzy $C$ -means (BPFCMs) algorithm is employed to define the hidden-layer nodes, extending traditional FCM through non-Gaussian modeling and posterior inference via Markov chain Monte Carlo (MCMC). By combining Metropolis-Hastings (MHs) for membership updates with Gibbs sampling for parameter estimation, BPFCM yields probabilistic memberships that capture uncertainty in the antecedent rules more effectively than deterministic approaches. Since the hidden-layer activations represent only similarity values between inputs and cluster centers, the original input features are not directly preserved. To compensate, the hidden-to-output connections are formulated as linear functions of the input, ensuring recovery of discriminative information in the consequent rules. These functions are optimized using a generalized cross-entropy (GCE) objective, with iteratively reweighted least squares (IRLSs) employed for efficient and regularized updates. Extensive experiments on benchmark datasets and high-dimensional laser-induced breakdown spectroscopy (LIBS) spectral data confirm that BPFNN consistently surpasses both classical fuzzy systems and contemporary deep learning models, providing improved accuracy, robustness, and interpretability.