For some one-dimensional input data ( \(x\) ), generate a real-valued latent function using a GP prior with an RBF covariance function. For each value, sample a random integer from a Poisson distribution with rate \(\lambda=\exp (f)\), i.e. the probability of getting the value \(z\) is equal to

\[P(z)=\frac{\lambda^{z} \exp (-\lambda)}{z!}\]

From this data, we would like to reverse-engineer the latent function. Compute the gradient and Hessian required to find the MAP solution to \(\mathbf{f}\) and compare the inferred value to the true value from which the data were generated.