# How to select noninformative priors on hyperparameters in gaussian process regression

I want to do a full Bayesian analysis of a Gaussian process regression problem: $$x\sim{GP[\mu,K(t_i,t_j)]},$$ where $\mu$ and $K$ are mean and covariance functions and $$K(t_i,t_j)=\sigma_f^2\exp\left(-\frac{(t_i-t_j)^2}{2l^2}\right)+\sigma_n^2\delta(t_i,t_j),$$ where $\sigma_f^2$,$l$, and $\sigma_n^2$ are parameters of the covariance function. It seems like I can select a noninformative prior density on $\mu$ to be equal to $1$. How do I select noninformative hyperpriors for $\sigma_f^2$,$l$, and $\sigma_n^2$? Any reference will be very helpful.

Having said that, you could use proper uniform priors on the covariance and noise parameters, for example with a lower limit of 0 and an upper limit of $U$. Then you could vary $U$ to ensure that the arbitrary upper limit is not unduly affecting your results.