We propose a probabilistic extension of Wiener-Laguerre models for causal operator learning. Classical Wiener-Laguerre models parameterize stable linear dynamics using orthonormal Laguerre bases and apply a static nonlinear map to the resulting features. While structurally efficient and interpretable, they provide only deterministic point estimates. We reinterpret the nonlinear component through the lens of Barron function approximation, viewing two-layer networks, random Fourier features, and extreme learning machines as discretizations of integral representations over parameter measures. This perspective naturally admits Bayesian inference on the nonlinear map and yields posterior predictive uncertainty. By combining Laguerre-parameterized causal dynamics with probabilistic Barron-type nonlinear approximators, we obtain a structured yet expressive class of causal operators equipped with uncertainty quantification. The resulting framework bridges classical system identification and modern measure-based function approximation, providing a principled approach to time-series modeling and nonlinear systems identification.

翻译：我们提出了一种用于因果算子学习的 Wiener-Laguerre 模型的概率扩展。经典的 Wiener-Laguerre 模型使用正交归一化的 Laguerre 基对稳定线性动力学进行参数化，并对生成的特征应用静态非线性映射。虽然结构高效且可解释，但它们仅提供确定性的点估计。我们通过 Barron 函数逼近的视角重新解释非线性分量，将两层网络、随机傅里叶特征和极限学习机视为对参数测度上积分表示的离散化。这一视角自然地允许对非线性映射进行贝叶斯推断，并产生后验预测不确定性。通过将 Laguerre 参数化的因果动力学与概率型 Barron 类非线性逼近器相结合，我们获得了一类结构化且富有表现力的因果算子，并配备了不确定性量化能力。所得框架连接了经典的系统辨识与现代的基于测度的函数逼近，为时间序列建模和非线性系统辨识提供了一种原则性方法。