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Most normative models in computational neuroscience describe the task of learning as the optimisation of a cost function with respect to a set of parameters. However, learning as optimisation fails to account for a time varying environment during the learning process; and the resulting point estimate in parameter space does not account for uncertainty. Here, we frame learning as filtering, i.e., a principled method for including time and parameter uncertainty. We derive the filtering-based learning rule for a spiking neuronal network - the Synaptic Filter - and show its computational and biological relevance. For the computational relevance, we show that filtering in combination with Bayesian regression improves performance compared to a gradient learning rule with optimal learning rate in terms of weight estimation. Furthermore, the filtering-based rule outperforms gradient-based rules in the presence of model mismatch, indicating a better generalisation performance. The dynamics of the mean of the Synaptic Filter is consistent with the spike-timing dependent plasticity (STDP) while the dynamics of the variance makes novel predictions regarding spike-timing dependent changes of EPSP variability. Moreover, the Synaptic Filter explains experimentally observed negative correlations between homo- and heterosynaptic plasticity.