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  1. A broad class of stochastic volatility models are defined by systems of stochastic differential equations, and while these models have seen widespread success in domains such as finance and statistical climatology, they typically lack an ability to condition on historical data to produce a true posterior distribution. To address this fundamental limitation, we show how to re-cast a class of stochastic volatility models as a hierarchical Gaussian process (GP) model with specialized covariance functions. This GP model retains the inductive biases of the stochastic volatility model while providing the posterior predictive distribution given by GP inference. Within this framework, we take inspiration from well studied domains to introduce a new class of models, Volt and Magpie, that significantly outperform baselines in stock and wind speed forecasting, and naturally extend to the multitask setting. 
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  2. Models such as convolutional neural networks restrict the hypothesis space to a set of functions satisfying equivariance constraints, and improve generalization in problems by capturing relevant symmetries. However, symmetries are often only partially respected, preventing models with restriction biases from fitting the data. We introduce Residual Pathway Priors (RPPs) as a method for converting hard architectural constraints into soft priors, guiding models towards structured solutions while retaining the ability to capture additional complexity. RPPs are resilient to approximate or misspecified symmetries, and are as effective as fully constrained models even when symmetries are exact. We show that RPPs provide compelling performance on both model-free and model-based reinforcement learning problems, where contact forces and directional rewards violate the assumptions of equivariant networks. Finally, we demonstrate that RPPs have broad applicability, including dynamical systems, regression, and classification. 
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