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Quantifying biomechanical properties of the human vasculature could deepen our understanding of cardiovascular diseases. Standard nonlinear regression in constitutive modeling requires considerable high-quality data and an explicit form of the constitutive model as prior knowledge. By contrast, we propose a novel approach that combines generative deep learning with Bayesian inference to efficiently infer families of constitutive relationships in data-sparse regimes. Inspired by the concept of functional priors, we develop a generative adversarial network (GAN) that incorporates a neural operator as the generator and a fully-connected neural network as the discriminator. The generator takes a vector of noise conditioned on measurement data as input and yields the predicted constitutive relationship, which is scrutinized by the discriminator in the following step. We demonstrate that this framework can accurately estimate means and standard deviations of the constitutive relationships of the murine aorta using data collected either from model-generated synthetic data or ex vivo experiments for mice with genetic deficiencies. In addition, the framework learns priors of constitutive models without explicitly knowing their functional form, providing a new model-agnostic approach to learning hidden constitutive behaviors from data. [ABSTRACT FROM AUTHOR]