code: def backward(
      ctx: FunctionCtx, dout_dmu: Tensor, dout_dL: Tensor
  ) -> Tuple[Tensor, Tensor]:
      mu, L, C = ctx.saved_tensors
      dout_dnat1, dout_dnat2 = _NaturalToMuVarSqrt._backward(
          dout_dmu, dout_dL, mu, L, C
      )
      """
          Now we need to do the Jacobian-Vector Product for the transformation:
          L = inv(chol(inv(-2 theta_cov)))
  
          C^T C = -2 theta_cov
  
          so we need to do forward differentiation, starting with sensitivity (sensitivities marked with .dots.)
          .theta_cov. = dout_dnat2
  
          and ending with sensitivity .C.
  
          if B = inv(-2 theta_cov) then:
  
          .B.  =  d inv(-2 theta_cov)/dtheta_cov * .theta_cov.  =  -B (-2 .theta_cov.) B
  
          if L = chol(B), B = LL^T then (https://homepages.inf.ed.ac.uk/imurray2/pub/16choldiff/choldiff.pdf):
  
          .L. = L phi(L^{-1} .B. (L^{-1})^T) = L phi(2 L^T .theta_cov. L)
  
          Then C = inv(L), so
  
          .C. = -C .L. C = phi(-2 L^T .theta_cov. L)C
          """
      A = L.transpose(-2, -1) @ dout_dnat2 @ L
      phi = _phi_for_cholesky_(A.mul_(-2))
      dout_dtril = phi @ C
      return dout_dnat1, dout_dtril
  
file_path: gpytorch/gpytorch/variational/tril_natural_variational_distribution.py
imports: from typing import Tuple
  import torch
  from linear_operator.operators import CholLinearOperator, TriangularLinearOperator
  from torch import Tensor
  from torch.autograd.function import FunctionCtx
  from ..distributions import Distribution, MultivariateNormal
  from .natural_variational_distribution import _NaturalToMuVarSqrt, _NaturalVariationalDistribution, _phi_for_cholesky_, _triangular_inverse
label_desc: Use image node_id 3 for calling the _TrilNaturalToMuVarSqrt obj's underlying member method code with example usage: obj.backward(ctx, dout_dmu, dout_dL) and returns: dout_dnat1,  dout_dtril