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The problem of stochastically robust minimum mean square error (MMSE) transceiver design is addressed for multiple-input multiple-output (MIMO) point-to-point channels with different imperfect channel state information (CSI) at the receiver and the transmitter. While the receiver has distribution knowledge of the doubly correlated Gaussian channel that is conditioned on pilot-based training observations (partial CSIRx), the transmitter has either conditional distribution knowledge about the receiver's observations based on feedback (partial CSITx), or only unconditioned distribution knowledge (statistical CSITx). In case of partial CSITx, the design is based on an alternating optimization of the transmit and receive filter. For statistical CSITx, a novel closed-form expression for the expected MMSE is calculated and the structure of the optimal precoder is determined. This enables us to employ an efficient gradient projection method for the robust precoder design.