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We present a proof-of-concept for controlling the grasp of an anthropomorphic mechatronic prosthetic hand by using a biomimetic tactile sensor, Bayesian inference, and simple algorithms for estimation and control. The sensor takes advantage of its compliant mechanics to provide a triaxial force sensing end-effector for grasp control. By calculating normal and shear forces at the fingertip, the prosthetic hand is able to maintain perturbed objects within the force cone to prevent slip. A Kalman filter is used as a noise-robust method to calculate tangential forces. Biologically inspired algorithms and heuristics are presented that can be implemented online to support rapid, reflexive adjustments of grip.