def _calculate_acceptance_probabilities(init_probs, target_probs):
  """Calculate the per-class acceptance rates.

  Args:
    init_probs: The class probabilities of the data.
    target_probs: The desired class proportion in minibatches.
  Returns:
    A list of the per-class acceptance probabilities.

  This method is based on solving the following analysis:

  Let F be the probability of a rejection (on any example).
  Let p_i be the proportion of examples in the data in class i (init_probs)
  Let a_i is the rate the rejection sampler should *accept* class i
  Let t_i is the target proportion in the minibatches for class i (target_probs)

F = sum_i(p_i (1-a_i)) = 1 - sum_i(p_i a_i) using sum_i(p_i) = 1

An example with class `i` will be accepted if `k` rejections occur, then an
  example with class `i` is seen by the rejector, and it is accepted. This can
  be written as follows:

t_i = sum_k=0^inf(F^k p_i a_i) = p_i a_j / (1 - F) using geometric series identity, since 0 <= F < 1 = p_i a_i / sum_j(p_j * a_j) using F from above

Note that the following constraints hold:

0 <= p_i <= 1, sum_i(p_i) = 1 0 <= a_i <= 1 0 <= t_i <= 1, sum_i(t_i) = 1

A solution for a_i in terms of the other variabes is the following:
    ```a_i = (t_i / p_i) / max_i[t_i / p_i]

"""

Make list of t_i / p_i.

ratio_l = target_probs / init_probs

Replace NaNs with 0s.

ratio_l = math_ops.select(math_ops.is_nan(ratio_l), array_ops.zeros_like(ratio_l), ratio_l)

Calculate list of acceptance probabilities.

max_ratio = math_ops.reduce_max(ratio_l) return ratio_l / max_ratio ```