The Law of Total Odds

TL;DR

This paper introduces the law of total odds as a more conceptually sound and unbiased alternative to the law of total probability for estimating class probabilities in binary classification, especially under dataset shift.

Contribution

It proposes the law of total odds, demonstrating its unbiasedness and connection to maximum likelihood estimation, improving probability estimation under dataset shift.

Findings

Total odds estimator is unbiased.

Sample total odds estimator coincides with a known maximum-likelihood estimator.

Law of total odds can transform conditional probabilities with known unconditional class probabilities.

Abstract

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.