MEMS gyroscopes are gaining popularity because of their low manufacturing costs in large quantities. For navigation system engineering, this presents a challenge because of strong nonstationary noise processes, such as 1/f noise, in the output of MEMS gyros. In practice, on-the-fly calibration is often required before the gyroscope data are useful and comparable to more expensive optical gyroscopes. In this paper, we focus on an important part of MEMS gyro processing, i.e., predicting the future bias given calibration data with known (usually zero) input. We derive prediction algorithms based on Kalman filtering and the computation of moving averages, and compare their performance against simple averaging of the calibration data based on both simulations and real measured data. The results show that it is necessary to model fractional noise in order to consistently predict the bias of a modern MEMS gyro, but the complexity of the Kalman filter approach makes other methods, such as the moving averages, appealing.