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.
Statement of the rule of succession 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,
P ( X n + 1 = 1 ∣ X 1 + ⋯ + X n = s ) = s + 1 n + 2 . {\displaystyle P(X_{n+1}=1\mid X_{1}+\cdots +X_{n}=s)={s+1 \over n+2}.}
Interpretation 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
P ′ ( X n + 1 = 1 ∣ X 1 + ⋯ + X n = s ) = s n . {\displaystyle P'(X_{n+1}=1\mid X_{1}+\cdots +X_{n}=s)={s \over n}.}
But see Mathematical details, below, for an analysis of its validity. In particular it is not valid when s = 0 {\displaystyle s=0} , or s = n {\displaystyle s=n} . If the number of observations increases, P {\displaystyle P} and P ′ {\displaystyle P'} get more and more similar, which is intuitively clear: the more data we have, the less importance should be assigned to our prior information.
Historical application to the sunrise problem 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. However, as the mathematical details below show, the basic assumption for using the rule of succession would be that we have no prior knowledge about the question whether the Sun will or will not rise tomorrow, except that it can do either. This is not the case for sunrises. Laplace knew this well, and he wrote to conclude the sunrise example: "But this number is far greater for him who, seeing in the totality of phenomena the principle regulating the days and seasons, realizes that nothing at the present moment can arrest the course of it." Yet Laplace was ridiculed for this calculation; his opponents gave no heed to that sentence, or failed to understand its importance. In the 1940s, Rudolf Carnap investigated a probability-based theory of inductive reasoning, and developed measures of degree of confirmation, which he considered as alternatives to Laplace's rule of succession. See also New riddle of induction#Carnap.
Intuition
… excerpt ends here. Continue reading the full article.
