Modification of synaptic connections is thought to be the core mechanism of long-term memory formation. Despite the recent experimental development, there are still challenges inferring the synaptic plasticity rule from the experimental data. Here, we considered a biologically plausible synaptic plasticity model, and explored its key properties that enable efficient inference. As a recent theoretical work suggested the firing rates to be the most critical under in vivo-like conditions, we considered firing-rate dependent plasticity and its matrix representation. To enable efficient recovery under a low sampling ratio, we enforced the smoothness property of the plasticity rule and utilized a Bayesian approach, known as Gaussian process regression. We first considered in-vitro experiments where the changes of synaptic weights can be measured directly for a few pairs of pre- and post-synaptic rates. We found that this Bayesian method outperforms alternative methods assuming low rankness of the matrix and smoothness utilizing polynomial or Fourier basis functions under variation of learning parameters that generate different plasticity rules. For in-vivo experiments where the direct measurement of synaptic weights is limited, inference of synaptic plasticity rule from changes of network activity with learning could be reformulated as a matrix completion problem with affine constraints. Compared to the previous work showing a partial inference from a single cell recording obtained in vivo, population recording and the Bayesian approach allow the inference of a complete learning rule. Our method performs stably under experimental restrictions such as input noise and missing synaptic connections. Also, we could generalize the method to the case when the change of post-synaptic activity reflects a mixture of different plasticity, such as excitatory and inhibitory plasticity. Overall, we introduced a non-parametric and model-agnostic method for efficient synaptic learning rule inference that can apply to both in-vitro and in-vivo experimental settings.