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tokens: much shorter or much longer than the Galactic orbital timescales, the distributions of element abundances at any orbital phase should not be affected by these subtleties. We therefore expect this to be inconsequential. Finally, we have assumed all along that there are measurable gradients in stellar abundances with respect to kinematics in the Galaxy or stellar population of interest. In the Milky Way, these are readily observed with existing spectroscopic surveys, and our measurements of individual element abundances will only become more numerous and precise in the coming years. 7.4. Extensions of Orbital Torus Imaging The implementation of Orbital Torus Imaging used in this paper falls into the category of classical (frequentist) statistics. However, the concepts introduced here also lead naturally to probabilistic generative model (Bayesian) versions of Orbital Torus Imaging. Bayesian implementations of this method at first look very different: instead of trying to minimize the dependence of abundance-ratio statistics on the conjugate angles (as we do here), Bayesian formulations would instead involve building a flexible forward model for the element-abundance distribution as a function of the actions alone. A non-intuitive aspect of this is that the angles would never explicitly appear in the probabilistic model. The Bayesian implementations would therefore look conceptually like extended DFs (Sanders Binney 2015 ), but while also varying the mass models, or forward models of the DF (e.g., Magorrian 2014 ), where element abundances or stellar labels are treated as additional invariants. In the implementation presented here, we use an abundance deviation that is based on a nearest-neighbor interpolation in action space (see Section 6 and Appendix ). That is a blunt tool; in principle, our results would improve if we instead explicitly modeled the abundance distribution moments as functions of the actions. One option for this modeling would be to use a generative, physical model for the star formation history of the Galaxy (or sample), including the products of nucleosynthesis and effects of chemical enrichment and stellar migration (similar to what is done in Sanders Binney 2015 ). Another option would be to use a flexible machine-learning method, such as a Gaussian process. We have parameterized the Milky Way mass model with a very simple, few-component mass model. Another option would be to instead parameterize the torus foliation of phase space directly. The representation of this foliation could be very general: any foliation of phase space with closed 3-tori is, in principle, allowed by dynamics. A direct reconstruction of the foliation could then be interpreted in terms of the force law, and hence the potential or mass model. That would be a more data-driven approach—with far fewer assumptions—than what has been executed here. We used the means of a set of hand-chosen abundance-ratio deviations as our invariants for the inferences.