Locally weighted regression (LWR) was created as a nonparametric method that can approximate a wide range of functions, is computationally efficient, and can learn continually from very large amounts of incrementally collected data. As an interesting feature, LWR can regress on non-stationary functions, a beneficial property, for instance, in control problems. However, it does not provide a proper generative model for function values, and existing algorithms have a variety of manual tuning parameters that strongly influence bias, variance and learning speed of the results. Gaussian (process) regression, on the other hand, does provide a generative model with rather black-box automatic parameter tuning, but it has higher computational cost, especially for big data sets and if a non-stationary model is required. In this paper, we suggest a path from Gaussian (process) regression to locally weighted regression, where we retain the best of both approaches. Using a localizing function basis and approximate inference techniques, we build a Gaussian (process) regression algorithm of increasingly local nature and similar computational complexity to LWR. Empirical evaluations are performed on several synthetic and real robot datasets of increasing complexity and (big) data scale, and demonstrate that we consistently achieve on par or superior performance compared to current state-of-the-art methods while retaining a principled approach to fast incremental regression with minimal manual tuning parameters.