Inverse problems are prevalent across various fields of science and engineering, with applications spanning medical imaging, materials science, non-destructive testing, astrophysics, climate science, and seismology. These problems share a common goal: estimating a quantity of interest from measurements obtained under specific experimental conditions. Traditionally, simulation and inference have been considered separate tasks. However, recent advancements in conditional variational inference offer a promising approach to integrate these tasks within a single framework. This paper reviews some of these advancements in the context of linear inverse problems, focusing on the use of straightforward generative models for inference and experimental design in computed tomography.