# Where does the scale of the priors recommended in Stan Users' Guide for Gaussian Processes come from?

In the Stan Users’ Guide (section 11.3, page 140), it suggests the following priors for a Gaussian process:

\rho \sim \operatorname{InvGamma}(5,5)
\alpha \sim \operatorname{Normal}(0,1)
\sigma \sim \operatorname{Normal}(0,1)

I have a rough idea for why the priors are as they are, but one thing that isn’t quite clear to me is the reason for particular values of the arguments of \operatorname{InvGamma} and \operatorname{Normal} (or really half-normal) distributions. I’m guessing that the standard deviations of the normal distributions are a rough estimate of the expected ranges of values for \alpha and \sigma, but I’m far more unsure of the reasoning behind why the arguments of the inverse gamma distribution are what they are, and I’m especially unsure what would change when the expected size of the length scale changes. Would, for example, the first argument of \operatorname{InvGamma} stay the same in order to reflect the shape of the left tail, while the second argument is estimated from a guestimate of the mean or mode of the distribution?

The \alpha parameter is the marginal standard deviation of the GP and should be set like any standard deviation parameter. Something like a half-Normal(0,1) or a half-normal(0,5) or and Exponential with an appropriate mean or a half-t with 3-7 degrees of freedom.