AbstractEquality‐constrained models naturally arise in problems in which the measurements are taken at different levels of resolution. The challenge in this setting is that the models usually induce a joint distribution which is intractable. Resorting to instead sampling from the joint distribution by means of a Monte Carlo approach is also challenging. For example, a naive rejection sampler does not work when the probability mass of the constraint is zero. A typical example of such constrained problems is to learn energy consumption for a higher resolution level based on data at a lower resolution, for example, to decompose a daily reading into readings at a finer level. We introduce a novel Monte Carlo sampling algorithm based on Langevin diffusions and rejection sampling to solve the problem of sampling from equality‐constrained models. Our method has the advantage of being exact for linear constraints and naturally deals with multimodal distributions on arbitrary constraints. We test our method on statistical disaggregation problems for electricity consumption datasets, and our approach provides better uncertainty estimation and accuracy in data imputation compared with other naive/unconstrained methods.