code: def backward(ctx, grad_output):
      (
          z,
          coord_scale,
          component_logits,
          component_scale,
          pis,
          coeffs,
      ) = ctx.saved_tensors
      dim = coord_scale.size(0)
      g = grad_output  # l i
      g = g.unsqueeze(-2)  # l 1 i
  
      component_scale_sqr = torch.pow(component_scale, 2.0)  # j
      epsilons = z / coord_scale  # l i
      epsilons_sqr = torch.pow(epsilons, 2.0)  # l i
      r_sqr = epsilons_sqr.sum(-1, keepdim=True)  # l
      r_sqr_j = r_sqr / component_scale_sqr  # l j
      coord_scale_product = coord_scale.prod()
      component_scale_power = torch.pow(component_scale, float(dim))
  
      q_j = torch.exp(-0.5 * r_sqr_j) / math.pow(
          2.0 * math.pi, 0.5 * float(dim)
      )  # l j
      q_j /= coord_scale_product * component_scale_power  # l j
      q_tot = (pis * q_j).sum(-1, keepdim=True)  # l
  
      Phi_j = torch.exp(-0.5 * r_sqr_j)  # l j
      exponents = -torch.arange(1.0, int(dim / 2) + 1.0, 1.0)
      if z.dim() > 1:
          r_j_poly = r_sqr_j.unsqueeze(-1).expand(
              -1, -1, int(dim / 2)
          )  # l j d/2
      else:
          r_j_poly = r_sqr_j.unsqueeze(-1).expand(
              -1, int(dim / 2)
          )  # l j d/2
      r_j_poly = coeffs * torch.pow(r_j_poly, exponents)
      Phi_j *= r_j_poly.sum(-1)
      if dim % 2 == 1:
          root_two = math.sqrt(2.0)
          extra_term = (
              coeffs[-1]
              * math.sqrt(0.5 * math.pi)
              * (1.0 - torch.erf(r_sqr_j.sqrt() / root_two))
          )  # l j
          Phi_j += extra_term * torch.pow(r_sqr_j, -0.5 * float(dim))
  
      logits_grad = (z.unsqueeze(-2) * Phi_j.unsqueeze(-1) * g).sum(
          -1
      )  # l j
      logits_grad /= q_tot
      logits_grad = sum_leftmost(logits_grad, -1) * math.pow(
          2.0 * math.pi, -0.5 * float(dim)
      )
      logits_grad = (
          pis
          * logits_grad
          / (component_scale_power * coord_scale_product)
      )
      logits_grad = logits_grad - logits_grad.sum() * pis
  
      prefactor = (
          pis.unsqueeze(-1)
          * q_j.unsqueeze(-1)
          * g
          / q_tot.unsqueeze(-1)
      )  # l j i
      coord_scale_grad = sum_leftmost(
          prefactor * epsilons.unsqueeze(-2), -1
      )
      component_scale_grad = sum_leftmost(
          (prefactor * z.unsqueeze(-2)).sum(-1) / component_scale, -1
      )
  
      return (
          coord_scale_grad,
          logits_grad,
          component_scale_grad,
          None,
          None,
          None,
          None,
      )
  
file_path: pyro/pyro/distributions/gaussian_scale_mixture.py
imports: import math
  import torch
  from torch.autograd import Function
  from torch.autograd.function import once_differentiable
  from torch.distributions import Categorical, constraints
  from pyro.distributions.torch_distribution import TorchDistribution
  from pyro.distributions.util import sum_leftmost
label_desc: Use image node_id 2 for calling the _GSMSample obj's underlying member method code with example usage: obj.backward(ctx, grad_output) and returns: coord_scale_grad,  logits_grad,  component_scale_grad,  None,  None,  None,  None