The objective optimization of medical imaging systems requires full characterization of all sources of randomness in the measured data, which includes the variability within the ensemble of objects to-be-imaged. This can be accomplished by establishing a stochastic object model (SOM) that describes the variability in the class of objects to-be-imaged. Generative adversarial networks (GANs) can be potentially useful to establish SOMs because they hold great promise to learn generative models that describe the variability within an ensemble of training data. However, because medical imaging systems record imaging measurements that are noisy and indirect representations of object properties, GANs cannot be directly applied to establish stochastic models of objects to-be-imaged. To address this issue, an augmented GAN architecture named AmbientGAN was developed to establish SOMs from noisy and indirect measurement data. However, because the adversarial training can be unstable, the applicability of the AmbientGAN can be potentially limited. In this work, we propose a novel training strategy---Progressive Growing of AmbientGANs (ProAGAN)---to stabilize the training of AmbientGANs for establishing SOMs from noisy and indirect imaging measurements. An idealized magnetic resonance (MR) imaging system and clinical MR brain images are considered. The proposed methodology is evaluated by comparing signal detection performance computed by use of ProAGAN-generated synthetic images and images that depict the true object properties.