In this last chapter, we discuss a final theoretical step of the moments approach illustrated in Chapter 13: under the assumption that the macroscopic profiles (e.g., density and velocity dispersion) of each component are known, there is a possibility of recovering the phase-space distribution function (DF) of a model and checking its positivity (i.e., verifying the model consistency). The problem of recovering the DF is in general a technically difficult inverse problem, and even when it is doable, unicity of the recovered DF is not guaranteed, so that a simple consistency analysis is quite problematic. Fortunately, there are special cases when (in principle) the DF can be obtained analytically (generally in integral form), and in these cases a few general and useful consistency conditions can be proved, such as the so-called global density slope–anisotropy inequality. The student is warned that this chapter is somewhat more technical than the others; however, the additional effort needed for its study will be well repaid by the understanding of some nontrivial results allowing for the construction of phase-space consistent collisionless stellar systems.