Galaxy morphology is connected to various fundamental properties of a galaxy and studying the morphology of large samples of galaxies is central to understanding the relationship between morphology and the physics of galaxy formation and evolution. For the first time, we are able to use machine learning to estimate Bayesian posteriors for galaxy morphological parameters. To achieve this, GAMPEN, our machine learning framework, uses a spatial transformer network (STN), a convolutional neural network, and the Monte-Carlo Dropout technique. This novel application of an STN in astronomy also enables GAMPEN to crop out most secondary galaxies in the frame and focus on the galaxy of interest. We also demonstrate that by first training on simulations and then performing transfer learning using real data, we are able to achieve excellent estimates for morphological parameters of galaxies in the Hyper Suprime-Cam Wide survey, while using only a small amount of real training data.