We introduce a novel machine learning framework for estimating the Bayesian posteriors of morphological parameters for arbitrarily large numbers of galaxies. The Galaxy Morphology Posterior Estimation Network (GaMPEN) estimates values and uncertainties for a galaxy's bulge-to-total light ratio, effective radius, and flux. GaMPEN also uses a Spatial Transformer Network (STN) to automatically crop input galaxy frames to an optimal size before determining their morphology. Training and testing GaMPEN on galaxies simulated to match z < 0.75 galaxies in Hyper Suprime-Cam Wide images, we demonstrate that GaMPEN can accurately quantify uncertainties and estimate parameters. GaMPEN is the first machine learning framework for determining posterior distributions of multiple morphological parameters and is also the first application of an STN to optical imaging in astronomy.