We present research on using machine learning and Bayesian inference to estimate parameters of the the Bayesian Observationally-constrained Statistical-physical Scheme (BOSS), a structurally flexible bulk warm microphysics scheme. We apply machine learning to act as a computationally efficient emulator of a full model's sensitivity to parameter perturbations, trained upon a perturbed parameter ensemble. This methodology has recently been used to develop candidates for the NASA GISS ModelE ESM, and others have also applied similar methodologies to E3SM and CESM (among others). We focus here on Large Eddy Simulations (LES) of warm shallow clouds and demonstrate that we can use pseudo-observations generated by LES runs with a reference bin scheme to perform Bayesian parameter estimation on BOSS within the same LES model. We then test these parameter values in E3SM, ModelE, and CESM, and show that they can produce realistic simulated climates. We discuss many critical issues related to this effort including how to estimate parameters in the presence of strong structural model errors, as well as the interplay of microphysical parameterization uncertainty with other model components such as the stratiform and convective cloud schemes. We discuss how observational uncertainty is accounted for, as well as how uncertainty in the machine learned emulator can be accounted for in our inference framework. Finally, we outline a workflow for combined bottom-up, top-down tuning of ESM physics, and remaining challenges that must be addressed.