The increasing integration of renewable energy requires a good model of the existing power distribution infrastructure, represented by its admittance matrix. However, a reliable estimate may either be missing or quickly become obsolete, as distribution grids are continuously modified. In this work, we propose a method for estimating the admittance matrix from voltage and current measurements. By focusing on $\mu $ PMU measurements and partially observed networks, we show that voltage collinearity and noisy samples of all electric variables are the main challenges for accurate identification. Moreover, the accuracy of maximum likelihood estimation is often insufficient in real-world scenarios. To overcome this problem, we develop a flexible Bayesian framework that allows one to exploit different forms of prior knowledge about individual line parameters, as well as network-wide characteristics such as the sparsity of the interconnections. Most importantly, we show how to use maximum likelihood estimates for tuning relevant hyperparameters, hence making the identification procedure self-contained. We also discuss numerical aspects of the maximum a posteriori estimate computation. Realistic simulations conducted on benchmark electrical systems demonstrate that, compared to other algorithms, our method can achieve significantly greater accuracy than previously developed methods.