Filling in the blanks. A method to infer the substructure membership and dynamics of 5D stars

In the solar neighbourhood, only $ of stars in the survey have a line-of-sight velocity ( contained within the RVS catalogue. These limitations restrict conventional dynamical analysis, such as finding and studying substructures in the stellar halo. We aim to present and test a method to infer a probability density function (PDF) for the missing of a star with 5D information within $2.5\ kpc $. This technique also allows us to infer the probability that a 5D star is associated with the Milky Way's stellar Disc or the stellar Halo, which can be further decomposed into known stellar substructures. We use stars from the DR3 RVS catalogue to describe the local orbital structure in action space. The method is tested on a 6D DR3 RVS sample and a 6D sample crossmatched to ground-based spectroscopic surveys, stripped of their true The stars predicted membership probabilities, and inferred structure properties are then compared to the true 6D equivalents, allowing the method's accuracy and limitations to be studied in detail. Our predicted PDFs are statistically consistent with the true with accurate uncertainties. We find that the of Disc stars can be well-constrained, with a median uncertainty of $26\ Halo stars are typically less well-constrained with a median uncertainty of $72\ but those found likely to belong to Halo substructures can be better constrained. The dynamical properties of the total sample and subgroups, such as distributions of integrals of motion and velocities, are also accurately recovered. The group membership probabilities are statistically consistent with our initial labelling, allowing high-quality sets to be selected from 5D samples by choosing a trade-off between higher expected purity and decreasing expected completeness. We have developed a method to estimate 5D stars' and substructure membership. We have demonstrated that it is possible to find likely substructure members and statistically infer the group's dynamical properties.