  In clustering problems, a central decision-maker is given a complete metric
graph over vertices and must provide a clustering of vertices that minimizes
some objective function. In fair clustering problems, vertices are endowed with
a color (e.g., membership in a group), and the features of a valid clustering
might also include the representation of colors in that clustering. Prior work
in fair clustering assumes complete knowledge of group membership. In this
paper, we generalize prior work by assuming imperfect knowledge of group
membership through probabilistic assignments. We present clustering algorithms
in this more general setting with approximation ratio guarantees. We also
address the problem of "metric membership", where different groups have a
notion of order and distance. Experiments are conducted using our proposed
algorithms as well as baselines to validate our approach and also surface
nuanced concerns when group membership is not known deterministically.
