Training-based estimation of channel state information in multi-antenna systems is analyzed herein. Closed-form expressions for the general Bayesian minimum mean square error (MMSE) estimators of the channel matrix and the squared channel norm are derived in a Rayleigh fading environment with known statistics at the receiver side. When the second-order channel statistics are available also at the transmitter, this information can be exploited in the training sequence design to improve the performance. Herein, mean square error (MSE) minimizing training sequences are considered. The structure of the general solution is developed, with explicit expressions at high and low SNRs and in the special case of uncorrelated receive antennas. The optimal length of the training sequence is equal or smaller than the number of transmit antennas.