The mobility of the thumb carpometacarpal (CMC) joint is critical for functional grasping and manipulation tasks. We present an optimization technique for determining from surface marker measurements a subject-specific kinematic model of the in vivo CMC joint that is suitable for measuring mobility. Our anatomy-based cost metric scores a candidate joint model by the plausibility of the corresponding joint angle values and kinematic parameters rather than only the marker trajectory reconstruction error. The proposed method repeatably determines CMC joint models with anatomically-plausible directions for the two dominant rotational axes and a lesser range of motion (RoM) for the third rotational axis. We formulate a low-dimensional parameterization of the optimization domain by first solving for joint axis orientation variables that then constrain the search for the joint axis location variables. Individual CMC joint models were determined for 24 subjects. The directions of the flexion--extension (FE) axis and adduction--abduction (AA) axis deviated on average by 9 degrees and 22 degrees , respectively, from the mean axis direction. The average RoM for FE, AA, and pronation--supination (PS) joint angles were 76 degrees , 43 degrees , and 23 degrees for active CMC movement. The mean separation distance between the FE and AA axes was 4.6 mm, and the mean skew angle was 87 degrees from the positive flexion axis to the positive abduction axis.