  The law of total probability may be deployed in binary classification
exercises to estimate the unconditional class probabilities if the class
proportions in the training set are not representative of the population class
proportions. We argue that this is not a conceptually sound approach and
suggest an alternative based on the new law of total odds. We quantify the bias
of the total probability estimator of the unconditional class probabilities and
show that the total odds estimator is unbiased. The sample version of the total
odds estimator is shown to coincide with a maximum-likelihood estimator known
from the literature. The law of total odds can also be used for transforming
the conditional class probabilities if independent estimates of the
unconditional class probabilities of the population are available.
  Keywords: Total probability, likelihood ratio, Bayes' formula, binary
classification, relative odds, unbiased estimator, supervised learning, dataset
shift.
