code: def inference(self, kern, X, likelihood, Y, mean_function=None, Y_metadata=None, K=None, variance=None, Z_tilde=None):
          
  
          if mean_function is None:
              m = 0
          else:
              m = mean_function.f(X)
  
          if variance is None:
              variance = likelihood.gaussian_variance(Y_metadata)
  
          YYT_factor = Y-m
  
          if K is None:
              K = kern.K(X)
  
          Ky = K.copy()
          diag.add(Ky, variance+1e-8)
  
          Wi, LW, LWi, W_logdet = pdinv(Ky)
  
          alpha, _ = dpotrs(LW, YYT_factor, lower=1)
  
          log_marginal =  0.5*(-Y.size * log_2_pi - Y.shape[1] * W_logdet - np.sum(alpha * YYT_factor))
  
          if Z_tilde is not None:
              # This is a correction term for the log marginal likelihood
              # In EP this is log Z_tilde, which is the difference between the
              # Gaussian marginal and Z_EP
              log_marginal += Z_tilde
  
          dL_dK = 0.5 * (tdot(alpha) - Y.shape[1] * Wi)
  
          dL_dthetaL = likelihood.exact_inference_gradients(np.diag(dL_dK), Y_metadata)
  
          return Posterior(woodbury_chol=LW, woodbury_vector=alpha, K=K), log_marginal, {'dL_dK':dL_dK, 'dL_dthetaL':dL_dthetaL, 'dL_dm':alpha}
docs: Returns a Posterior class containing essential quantities of the posterior