Here's the model in PyMC3. We use a Gamma(2,1) prior over the lengthscale parameter, and weakly informative HalfCauchy(5) priors over the covariance function scale, and noise scale. A Gamma(2,0.1) prior is assigned to the degrees of freedom parameter of the noise. Finally, a GP prior is placed on the unknown function.

https://nbviewer.jupyter.org/github/fonnesbeck/Bios366/blob/master/notebooks/Section5_1-Gaussian-Processes.ipynb