class_docstr: Pairwise likelihood using probit function
  
  Given two items v and u with utilities f(v) and f(u), the probability that we
  prefer v over u with probability std_normal_cdf((f(v) - f(u))/sqrt(2)). Note
  that this formulation implicitly assume the noise term is fixed at 1.
docstr: Calculate auxiliary statistics derived from z, including log pdf,
  log cdf, and the hazard function (pdf divided by cdf)
  
  Args:
      z: A Tensor of arbitrary shape.
  
  Returns:
      Tensors with standard normal logpdf(z), logcdf(z), and hazard function
      values evaluated at -z.
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code: def _calc_z_derived(self, z: Tensor) -> Tuple[Tensor, Tensor, Tensor]:
      """Calculate auxiliary statistics derived from z, including log pdf,
      log cdf, and the hazard function (pdf divided by cdf)
  
      Args:
          z: A Tensor of arbitrary shape.
  
      Returns:
          Tensors with standard normal logpdf(z), logcdf(z), and hazard function
          values evaluated at -z.
      """
      return (
          log_phi(z),
          log_ndtr(z),
          standard_normal_log_hazard(-z).exp(),
      )
  
imports: from __future__ import annotations
  import math
  from abc import ABC, abstractmethod
  from typing import Any, Tuple
  import torch
  from botorch.utils.probability.utils import log_ndtr, log_phi, standard_normal_log_hazard
  from gpytorch.likelihoods import Likelihood
  from torch import Tensor
  from torch.distributions import Bernoulli
label_desc: Use image node_id 2 for calling the PairwiseProbitLikelihood obj's underlying member method code with example usage: obj._calc_z_derived(z) and returns: log_phi,  log_ndtr,  standard_normal_log_hazard