This paper presents a noise adaptive variational Bayesian cubature information filter based on Wishart distribution. In the frame of recursive Bayesian estimation, the noise adaptive information filter propagating the information matrix and information state is derived. And the integration of recursive Bayesian estimation is approximated by cubature integration rule. Then, the inverse of measurement noise matrix is modeled as a Wishart distribution, so the joint distribution of posterior state and measurement noise can be approximated by the product of independent Gaussian and Wishart. Furthermore, the corresponding square root version is also derived to improve numerical characteristics. Simulation results with unknown and correlated measurement noise demonstrate the effectiveness of the proposed algorithms.