Consider two independent binomial distributions with probabilities of successes p_1 and p_2. If we observe a_1 successes, b_1 failures from the first distribution and a_2 successes, b_2 failures from the second distribution, what can we say about the difference, p_1 – p_2?
Binomial model differences like this were first studied by Laplace in 1778. Laplace observed that the ratio of boys-to-girls born in London was notably larger than the ratio of boys-to-girls born in Paris, and he sought to determine whether the difference was significant.
Using what would now be called Bayesian inference together with a uniform prior, Laplace computed the posterior probability that the birth ratio in London was less than the birth ratio in Paris as
where