This work addresses the design of optimal linear transmit filters for the Multiple Input-Multiple Output (MIMO) Broadcast Channel (BC) when several spatial streams are allocated to each user.We also consider that the Channel State Information (CSI) is perfect at the receivers but is only partial at the transmitter. A statistical model for the partial CSI is assumed and exploited for the filter design. Similarly to the single-stream per user case, the problem is solved via Mean Square Error (MSE) dualities and interference functions. However, including more streams per user involves an additional complexity level since we must determine how to distribute the per-user rates among the streams. Such problem is solved using a projected gradient algorithm.