Industrial robots are highly desirable in applications including manufacturing and surgery. However, errors in the modeling of the kinematics of robotic arms limit their positional accuracy in industrial applications. Specifically, analytical kinematic models of the robot arm suffer from errors in coefficient calibrations and the inability to account for effects including gear backlash. However, statistical modeling methods require an extensive amount of points for calibration, which is infeasible in practical industrial environments. Hence, this paper describes, develops, and experimentally validates a hybrid modeling methodology combining both analytical and statistical methods to describe the robot kinematics in an intuitive manner that is easily adaptable for small- and medium-sized industries. By formulating an explicitly described analytical kinematic model as a prior mean distribution of a Gaussian process, the prior distribution can be updated with experimental data using statistical Bayesian Inference, thus enabling more accurate description of the robot kinematics with fewer data points. The hybrid model is demonstrated to outperform an analytical model, a neural network model, and a Gaussian Process Regression model with no prior distribution in predicting both the forward and inverse kinematics of a UR5 and UR10 robot arm. Also, the error propagation of the inverse kinematic solutions is studied. In addition, the testing framework used in this work can be used as a standardized benchmark to evaluate alternative kinematic models.