Fisher Expected Information for a Gaussian Process model

Question

Suppose I have a two dimensional Gaussian process model (GP), defined by a squared exponential correlation function s.t: $$R(x_{i},x_{j}) = \exp\left(-\frac{|x_{i} - x_{j}|^2}{2}\right).$$ I am trying to evaluate Fisher Expected Information for the GP model, defined by: $$I(\theta) = E\left[\left(\frac{\partial \log f(x;\theta)}{d \theta}\right)^2\right].$$