Computational analysis of neural systems is at its most useful when it uncovers principles that provide a unified account of phenomena across multiple scales and levels of description. Here we analyse a widely used model of the cerebellar contribution to sensori-motor learning to demonstrate both that its response to intrinsic and sensor noise is optimal, and that the unexpected synaptic and behavioural consequences of this optimality can explain a wide range of experimental data. The response of the Marr-Albus adaptive-filter model of the cerebellar microcircuit to noise was examined in the context of vestibulo-ocular reflex calibration. We found that, when appropriately connected, an adaptive-filter model using the covariance learning rule to adjust the weights of synapses between parallel fibres and Purkinje cells learns weight values that are optimal given the relative amount of signal and noise carried by each parallel fibre. This optimality principle is consistent with data on the cerebellar role in smooth pursuit eye movements, and predicts that many synaptic weights must be very small, providing an explanation for the experimentally observed preponderance of silent synapses. Such a preponderance has in its turn two further consequences. First, an additional inhibitory pathway from parallel fibre to Purkinje cell is required if Purkinje cell activity is to be altered in either direction from a starting point of silent synapses. Second, cerebellar learning tasks must often proceed via LTP, rather than LTD as is widely assumed. Taken together, these considerations have profound behavioural consequences, including the optimal combination of sensori-motor information, and asymmetry and hysteresis of sensori-motor learning rates.