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.
 A: There is no such thing as a truly noninformative prior--even the prior you've selected for the mean is not uniform under a reparameterisation.
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.
However, I would generally advocate using an informative prior... but that is another answer.
A: Ren et al. derived reference priors (i.e. approximately noninformative priors) for Gaussian processes of this form in their paper Objective Bayesian Analysis for a Spatial Model with
Nugget Effects
Using the notation from their paper, let $\mathbf{y}$ denote the observations,
$\mathbf{s}_1, \ldots, \mathbf{s}_n$ denote the sampling locations, and
$$
\mathbf{X} = \left(\mathbf{x}(\mathbf{s}_1), \ldots, \mathbf{x}(s_n)\right)'
$$
denote the $n\times p$ design matrix of a linear mean function.
Then
$$
\mathbf{y} \sim N_n(\mathbf{X}\mathbf{\beta}, \delta_1 \mathbf{G}), \quad\mathbf{G} = \eta\mathbf{I}_n + \mathbf{K}(\theta)
$$
where $\delta_1$ is equivalent to $\sigma_f^2$ in your notation, $\eta$ is the noise-to-signal-ratio $\sigma_n^2/\sigma_f^2$, and $\theta$ represents the hyperparameters of the covariance function.
Then the reference prior $\pi^R_*$ (equation 23) from their paper is given by
$$
\pi^R_*(\theta, \eta, \delta_1, \beta) \propto \frac{1}{\delta_1} \left(\det\mathbf{\Sigma}_*(\theta, \eta)\right)^{1/2}
$$
where
$$
\mathbf{\Sigma}_*(\theta, \eta) = \frac{1}{2} 
   \begin{pmatrix}
      \mathrm{tr}\left[\left(\mathbf{R}_G \frac{\partial}{\partial \theta} 
\mathbf{K}(\theta)\right)^2\right] & 
\mathrm{tr}\left(\mathbf{R}_G^2 \frac{\partial}{\partial \theta} 
\mathbf{K}(\theta)\right) & 
\mathrm{tr}\left(\mathbf{R}_G \frac{\partial}{\partial \theta} 
\mathbf{K}(\theta)\right) \\
* & \mathrm{tr}\left(\mathbf{R}_G^2\right) & \mathrm{tr}\left(\mathbf{R}_G\right) \\
* & * & n - p \\
   \end{pmatrix}
$$
and
$$
\mathbf{R}_G = \mathbf{G}^{-1} - \mathbf{G}^{-1} \mathbf{X} \left(
   \mathbf{X}'\mathbf{G}^{-1}\mathbf{X}
   \right)^{-1} \mathbf{X}' \mathbf{G}^{-1}
$$
For additional references, see

*

*Background on reference priors: Berger et al. The formal definite of reference priors

*Noninformative priors in the simplified case where $\sigma_n=0$: Berger et al. Objective Bayesian Analysis of Spatially Correlated Data
