new_contents: from __future__ import division
  
  import numpy as np
  
  
  class DiffKernel(object):
      """
      A fake kernel that can be used to predict differences two function values.
  
      Given a gp based on measurements, we aim to predict the difference between
      the function values at two different test points, X1 and X2; that is, we
      want to obtain mean and variance of f(X1) - f(X1). Using this fake
      kernel, this can be achieved with
      `mean, var = gp.predict(np.vstack((X1, X2)), kern=DiffKernel(gp.kern))`
  
      Parameters
      ----------
      kernel: GPy.kern.*
          The kernel used by the GP
      """
  
      def __init__(self, kernel):
          self.kern = kernel
  
      def K(self, x1, x2=None):
          """Equivalent of kern.K
  
          If only x1 is passed then it is assumed to contain the data for both
          whose differences we are computing. Otherwise, x2 will contain these
          extended states (see PosteriorExact._raw_predict in
          GPy/inference/latent_function_inference0/posterior.py)
  
          Parameters
          ----------
          x1: np.array
          x2: np.array
          """
          dim = self.kern.input_dim
          if x2 is None:
              x10 = x1[:, :dim]
              x11 = x1[:, dim:]
              return (self.kern.K(x10) + self.kern.K(x11) -
                      2 * self.kern.K(x10, x11))
          else:
              x20 = x2[:, :dim]
              x21 = x2[:, dim:]
              return self.kern.K(x1, x20) - self.kern.K(x1, x21)
  
      def Kdiag(self, x):
          """Equivalent of kern.Kdiag for the difference prediction.
  
          Parameters
          ----------
          x: np.array
          """
          dim = self.kern.input_dim
          x0 = x[:, :dim]
          x1 = x[:, dim:]
          return (self.kern.Kdiag(x0) + self.kern.Kdiag(x1) -
                  2 * np.diag(self.kern.K(x0, x1)))
  
subject: Add fake kernel to predict function differences using GPy