How to implement Variational Gaussian Process with custom likelihood in gpflow

Question

I have this equation with , y is observation and f is latent. I write this new Likelihood to compute in my model:

class ExpGaussian(likelihoods.ScalarLikelihood):
  def Y_given_F(self, F):
    mu = tf.math.exp(F)
    sigma = tf.math.sqrt(4*mu)
    return tfd.Normal(mu, sigma)
  @inherit_check_shapes
  def _scalar_log_prob(self, X: TensorType, F: TensorType, Y: TensorType) -> tf.Tensor:
    # prob = tfd.Normal(tf.math.exp(F), 4*tf.math.exp(F))
    # return prob.log_prob(Y)
    return self.Y_given_F(F).log_prob(Y)
  @inherit_check_shapes
  def _conditional_mean(self, X: TensorType, F: TensorType) -> tf.Tensor:
    # return tf.math.exp(F)
    return self.Y_given_F(F).mean()
  @inherit_check_shapes
  def _conditional_variance(self, X: TensorType, F: TensorType) -> tf.Tensor:
    # return 4*tf.math.exp(F)
    return self.Y_given_F(F).variance()

My input X is [[0],[1],[2],...,[250]] and Y is its corresponding value, which looklike this After optimize, I predict value of mean Y in 20 steps ahead and got this: It's exponentially growth up! So is that normal or my model has any problem? Thanks a lot!

P/S: Here is colab link to my code https://colab.research.google.com/drive/1r9zpJhGwCV3qIybrJDC5SzZTurvEsfxG?usp=sharing