Bayesian Wavelet Regression for Multiscale Signal Modeling: Mathematical Foundations, Uncertainty-Aware Inference, and Applications in Neuroscience and Vision Science

Bayesian wavelet regression offers a mathematically grounded framework for modeling neuro-retinal signals that exhibit substantial noise, high dimensionality, and complex variations across temporal and spatial scales. Such characteristics are inherent to diverse neuroscience and vision science modalities, including electrophysiological recordings, neuroimaging, and retinal imaging, where accurate interpretation depends not only on effective signal representation but also on reliable uncertainty estimation and biologically meaningful feature identification. By combining wavelet-based multiresolution analysis with Bayesian probabilistic inference, these approaches provide adaptive mechanisms for separating informative structures from noise while maintaining clinically relevant signal characteristics. This review examines the mathematical principles and recent methodological advances in Bayesian wavelet regression, focusing on sparse signal representation, Bayesian regularization, and uncertainty-aware estimation. Particular emphasis is placed on sparsity-inducing prior formulations, including spike-and-slab, moment, and inverse moment priors, which enhance coefficient selection, improve parameter identifiability, and facilitate the recovery of informative multiscale patterns. In addition, Bayesian inference frameworks, including Markov Chain Monte Carlo, Hamiltonian Monte Carlo, No-U-Turn Sampling, and Variational Bayes, are analyzed with respect to posterior estimation accuracy, computational requirements, and practical applicability. The literature synthesis indicates that Bayesian wavelet approaches support multiscale representation, sparse coefficient selection, and probabilistic uncertainty characterization, with prior and inference choices involving trade-offs in sparsity, interpretability, estimation, and computational efficiency. These applications span neural signal analysis, neuroimaging, retinal imaging, and neuro-ophthalmology, while multimodal retina–brain integration and clinically validated uncertainty-aware modeling remain less developed. Extending beyond single-modality analysis, this review investigates hierarchical Bayesian models for multimodal brain–retina integration and compares Bayesian wavelet regression with Fourier methods, multitaper approaches, empirical mode decomposition, Gaussian processes, and deep learning. Using a hybrid systematic and narrative review approach guided by PRISMA principles, this work identifies challenges in computational scalability, cross-scale dependency modeling, multimodal uncertainty propagation, prior sensitivity, and clinical validation, and discusses emerging directions including Bayesian deep learning, graph wavelets, neural wavelet representations, foundation models, and real-time inference toward transparent, uncertainty-aware, and clinically meaningful neuro-retinal intelligence systems. In total, 237 selected studies are synthesized across methodological and application domains.