I will like to implement a fully Bayesian Gaussian process regression in Pyro. Below is the target function and the synthetic data.

```
# The target function:
def f(x):
return torch.sin(20*x) + 2*torch.cos(14*x) - 2*torch.sin(6*x)
# Generation of synthetic data set:
xs = torch.tensor([-1.,-0.5,0,0.5,1])
epsilon = torch.normal(0, 0.0004, size=5)
ys = f(xs) + epsilon
# The data set consists of xs and ys.
```

The kernel used is the RBF kernel:

The \sigma_{s}^{2} is a variance and the \sigma_{l}^{2} is a length scale. k\left(x,x^{\prime}\right) gives some measure of similarity. This is a fully Bayesian Gaussian process regression, so I choose the following priors on the parameters \sigma_{s}^{2} and \sigma_{l}^{2}: \sigma_{l}^{2} \sim \mathrm{Lognormal}(-1,1) and \sigma_{v}^{2} \sim \mathrm{Lognormal}(0,2).

How do I implement the above fully Bayesian regression model using the above data and kernel? Can you give me an example of a fully Bayesian GP regression in Pyro using NUTS to sample from the posterior using the above data and kernel?

For me the documentation is not sufficient, unfortunately. I have been working on this project for a long time with the goal of learning Pyro for GP regression—without any luck.