class_docstr: An alternative to the variational evidence lower bound (ELBO), proposed by `Knoblauch, 2019`_.
  It is derived by replacing the log-likelihood term in the ELBO with a `\gamma` divergence:
  
  .. math::
  
     \begin{align*}
        \mathcal{L}_{\gamma} &=
        \sum_{i=1}^N \mathbb{E}_{q( \mathbf u)} \left[
          -\frac{\gamma}{\gamma - 1}
          \frac{
              p( y_i \! \mid \! \mathbf u, x_i)^{\gamma - 1}
          }{
              \int p(y \mid \mathbf u, x_i)^{\gamma} \: dy
          }
        \right] - \beta \: \text{KL} \left[ q( \mathbf u) \Vert p( \mathbf u) \right]
     \end{align*}
  
  where :math:`N` is the number of datapoints, :math:`\gamma` is a hyperparameter,
  :math:`q(\mathbf u)` is the variational distribution for
  the inducing function values, and :math:`p(\mathbf u)` is the prior distribution for the inducing function
  values.
  
  :math:`\beta` is a scaling constant for the KL divergence.
  
  .. note::
      This module will only work with :obj:`~gpytorch.likelihoods.GaussianLikelihood`.
  
  :param ~gpytorch.likelihoods.GaussianLikelihood likelihood: The likelihood for the model
  :param ~gpytorch.models.ApproximateGP model: The approximate GP model
  :param int num_data: The total number of training data points (necessary for SGD)
  :param float beta: (optional, default=1.) A multiplicative factor for the KL divergence term.
      Setting it to anything less than 1 reduces the regularization effect of the model
      (similarly to what was proposed in `the beta-VAE paper`_).
  :param float gamma: (optional, default=1.03) The :math:`\gamma`-divergence hyperparameter.
  :param bool combine_terms: (default=True): Whether or not to sum the
      expected NLL with the KL terms (default True)
  
  Example:
      >>> # model is a gpytorch.models.ApproximateGP
      >>> # likelihood is a gpytorch.likelihoods.Likelihood
      >>> mll = gpytorch.mlls.GammaRobustVariationalELBO(likelihood, model, num_data=100, beta=0.5, gamma=1.03)
      >>>
      >>> output = model(train_x)
      >>> loss = -mll(output, train_y)
      >>> loss.backward()
  
  .. _Knoblauch, 2019:
      https://arxiv.org/pdf/1904.02303.pdf
  .. _Knoblauch, Jewson, Damoulas 2019:
      https://arxiv.org/pdf/1904.02063.pdf
code: def _log_likelihood_term(
      self, variational_dist_f, target, *args, **kwargs
  ):
      shifted_gamma = self.gamma - 1
  
      muf, varf = variational_dist_f.mean, variational_dist_f.variance
  
      # Get noise from likelihood
      noise = self.likelihood._shaped_noise_covar(
          muf.shape, *args, **kwargs
      ).diagonal(dim1=-1, dim2=-2)
      # Potentially reshape the noise to deal with the multitask case
      noise = noise.view(
          *noise.shape[:-1], *variational_dist_f.event_shape
      )
  
      # adapted from https://github.com/JeremiasKnoblauch/GVIPublic/
      mut = shifted_gamma * target / noise + muf / varf
      sigmat = 1.0 / (shifted_gamma / noise + 1.0 / varf)
      log_integral = -0.5 * shifted_gamma * torch.log(
          2.0 * math.pi * noise
      ) - 0.5 * np.log1p(shifted_gamma)
      log_tempered = (
          -math.log(shifted_gamma)
          - 0.5 * shifted_gamma * torch.log(2.0 * math.pi * noise)
          - 0.5 * torch.log1p(shifted_gamma * varf / noise)
          - 0.5 * (shifted_gamma * target.pow(2.0) / noise)
          - 0.5 * muf.pow(2.0) / varf
          + 0.5 * mut.pow(2.0) * sigmat
      )
  
      factor = log_tempered + shifted_gamma / self.gamma * log_integral
      factor = self.gamma * factor.exp()
  
      # Do appropriate summation for multitask Gaussian likelihoods
      num_event_dim = len(variational_dist_f.event_shape)
      if num_event_dim > 1:
          factor = factor.sum(list(range(-1, -num_event_dim, -1)))
  
      return factor.sum(-1)
  
file_path: gpytorch/gpytorch/mlls/gamma_robust_variational_elbo.py
imports: import math
  import numpy
  import torch
  from ..likelihoods import _GaussianLikelihoodBase
  from ._approximate_mll import _ApproximateMarginalLogLikelihood
label_desc: Use image node_id 2 for calling the GammaRobustVariationalELBO obj's underlying member method code with example usage: obj._log_likelihood_term(variational_dist_f, target) and returns: factor