Rule of succession

In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem. The formula is still used, particularly to estimate underlying probabilities when there are few observations or events that have not been observed to occur at all in (finite) sample data. If we repeat an experiment that we know can result in a success or failure, n times independently, and get s successes, and n − s failures, then what is the probability that the next repetition will succeed? More abstractly: If X1, ..., Xn+1 are conditionally independent random variables that each can assume the value 0 or 1, then, if we know nothing more about them, Since we have the prior knowledge that we are looking at an experiment for which both success and failure are possible, our estimate is as if we had observed one success and one failure for sure before we even started the experiments. In a sense we made n + 2 observations (known as pseudocounts) with s + 1 successes. Although this may seem the simplest and most reasonable assumption, which also happens to be true, it still requires a proof. Indeed, assuming a pseudocount of one per possibility is one way to generalise the binary result, but has unexpected consequences — see Generalization to any number of possibilities, below. Nevertheless, if we had not known from the start that both success and failure are possible, then we would have had to assign But see Mathematical details, below, for an analysis of its validity. In particular it is not valid when , or . If the number of observations increases, and get more and more similar, which is intuitively clear: the more data we have, the less importance should be assigned to our prior information. Laplace used the rule of succession to calculate the probability that the Sun will rise tomorrow, given that it has risen every day for the past 5000 years. One obtains a very large factor of approximately 5000 × 365.25, which gives odds of about 1,826,200 to 1 in favour of the Sun rising tomorrow.