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.Laplace, Pierre-Simon (1814). Essai philosophique sur les probabilités. Paris: Courcier. 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,
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
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.
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."Part II Section 18.6 of Jaynes, E. T. & Bretthorst, G. L. (2003). Probability Theory: The Logic of Science. Cambridge University Press. 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.; here: p.86, 97; here: p.145 See also New riddle of induction#Carnap.
Intuition
The rule of succession can be interpreted in an intuitive manner by considering points randomly distributed on a circle rather than counting the number "success"/"failures" in an experiment. To mimic the behavior of the proportion p on the circle, we will color the circle in two colors and the fraction of the circle colored in the "success" color will be equal to p. To express the uncertainty about the value of p, we need to select a fraction of the circle.
A fraction is chosen by selecting two uniformly random points on the circle. The first point Z corresponds to the zero in the [0, 1] interval and the second point P corresponds to p within [0, 1]. In terms of the circle the fraction of the circle from Z to P moving clockwise will be equal to p. The n trials can be interpreted as n points uniformly distributed on the circle; any point in the "success" fraction is a success and a failure otherwise. This provides an exact mapping from success/failure experiments with probability of success p to uniformly random points on the circle. In the figure the success fraction is colored blue to differentiate it from the rest of the circle and the points P and Z are highlighted in red.
Given this circle, the estimate of p is the fraction colored blue. Let us divide the circle into n+2 arcs corresponding to the n+2 points such that the portion from a point on the circle to the next point (moving clockwise) is one arc associated with the first point. Thus, Z defines the first blue arc while P defines the first non-blue/failure arc. Since the next point is a uniformly random point, if it falls in any of the blue arcs then the trial succeeds while if it falls in any of the other arcs, then it fails. So the probability of success p is where b is the number of blue arcs and t is the total number of arcs. Note that there is one more blue arc (that of Z) than success points and two more arcs (those of P and Z) than n points. Substituting the values with number of successes gives the rule of succession.
Note: The actual probability needs to use the length of blue arcs divided by the length of all arcs. However, when k points are uniformly randomly distributed on a circle, the distance from a point to the next point is 1/k. So on average each arc is of the same length and ratio of lengths becomes ratio of counts.
Mathematical details The proportion p is assigned a uniform distribution to describe the uncertainty about its true value. (This proportion is not random, but uncertain. We assign a probability distribution to p to express our uncertainty, not to attribute randomness to p. But this amounts, mathematically, to the same thing as treating p as if it were random).
Let Xi be 1 if we observe a "success" on the ith trial, otherwise 0, with probability p of success on each trial. Thus each X is 0 or 1; each X has a Bernoulli distribution. Suppose these Xs are conditionally independent given p.
We can use Bayes' theorem to find the conditional probability distribution of p given the data Xi, i = 1, ..., n. For the "prior" (i.e., marginal) probability measure of p we assigned a uniform distribution over the open interval (0,1)
For the likelihood of our observations under a given p, we use the likelihood function
where s = x1 + ... + xn is the number of "successes" and n is the number of trials (we are using capital X to denote a random variable and lower-case x as the data actually observed). Putting it all together, we can calculate the posterior:
To get the normalizing constant, we find
(see beta function for more on integrals of this form).
The posterior probability density function is therefore
This is a beta distribution with expected value
Since p tells us the probability of success in any experiment, and each experiment is conditionally independent, the conditional probability for success in the next experiment is just p. As p is being treated as if it is a random variable, the law of total probability tells us that the expected probability of success in the next experiment is just the expected value of p. Since p is conditional on the observed data Xi for i = 1, ..., n, we have
The same calculation can be performed with the (improper) prior that expresses total ignorance of p, including ignorance with regard to the question whether the experiment can succeed, or can fail. This improper prior is 1/(p(1 − p)) for 0 ≤ p ≤ 1 and 0 otherwise. If the calculation above is repeated with this prior, we get
Thus, with the prior specifying total ignorance, the probability of success is governed by the observed frequency of success. However, the posterior distribution that led to this result is the Beta(s,n − s) distribution, which is not proper when s = n or s = 0 (i.e. the normalisation constant is infinite when s = 0 or s = n). This means that we cannot use this form of the posterior distribution to calculate the probability of the next observation succeeding when s = 0 or s = n. This puts the information contained in the rule of succession in greater light: it can be thought of as expressing the prior assumption that if sampling was continued indefinitely, we would eventually observe at least one success, and at least one failure in the sample. The prior expressing total ignorance does not assume this knowledge.
To evaluate the "complete ignorance" case when s = 0 or s = n can be dealt with, we first go back to the hypergeometric distribution, denoted by . This is the approach taken in Jaynes (2003). The binomial can be derived as a limiting form, where in such a way that their ratio remains fixed. One can think of as the number of successes in the total population, of size .
The equivalent prior to is , with a domain of . Working conditional to means that estimating is equivalent to estimating , and then dividing this estimate by . The posterior for can be given as:
And it can be seen that, if s = n or s = 0, then one of the factorials in the numerator cancels exactly with one in the denominator. Taking the s = 0 case, we have:
Adding in the normalising constant, which is always finite (because there are no singularities in the range of the posterior, and there are a finite number of terms) gives:
So the posterior expectation for is:
An approximate analytical expression for large N is given by first making the approximation to the product term:
and then replacing the summation in the numerator with an integral
The same procedure is followed for the denominator, but the process is a bit more tricky, as the integral is harder to evaluate
where ln is the natural logarithm plugging in these approximations into the expectation gives
where the base 10 logarithm has been used in the final answer for ease of calculation. For instance if the population is of size 10k then probability of success on the next sample is given by:
So for example, if the population be on the order of tens of billions, so that k = 10, and we observe n = 10 results without success, then the expected proportion in the population is approximately 0.43%. If the population is smaller, so that n = 10, k = 5 (tens of thousands), the expected proportion rises to approximately 0.86%, and so on. Similarly, if the number of observations is smaller, so that n = 5, k = 10, the proportion rise to approximately 0.86% again.
This probability has no positive lower bound, and can be made arbitrarily small for larger and larger choices of N, or k. This means that the probability depends on the size of the population from which one is sampling. In passing to the limit of infinite N (for the simpler analytic properties) we are "throwing away" a piece of very important information. Note that this ignorance relationship only holds as long as only no successes are observed. It is correspondingly revised back to the observed frequency rule as soon as one success is observed. The corresponding results are found for the s=n case by switching labels, and then subtracting the probability from 1.
Generalization to any number of possibilities This section gives a heuristic derivation similar to that in Probability Theory: The Logic of Science.Jaynes, E.T. (2003), Probability Theory: The Logic of Science, Cambridge, UK, Cambridge University Press.