In daily life, we often face problems such as "statistics" and "small data". For example, what is the probability that the Ziqiang ship to Kaohsiung will be delayed? Is it possible that the brother-like baseball team, which is personally supported, will win in this end-of-season game? In fact, Pastor Bayes, who put forward the "Bayes' Theorem," gave some answers to similar questions.
Laplace's Law of Succession is a mathematical theory that explains the likelihood of an event occurring in a sample of n. It uses an example of a trial where a player will win or lose the game. If the person wins, the probability that they will win or lose is a certain percent. The probability that they will win the next time is a fixed percent of the chance that they will win.
This rule of succession is similar to Bayes' law of probability, but it is not the same as presidential or monarchical succession. A simple heuristic, the law of recurrence, can give a person a priori credence for an event before it actually happens. The more time passes, the less credence a person has for the occurrence.
The piquet problem, which is related to this law, was worked on by Pascal. In a piquet game, both players must estimate their relative skill levels, and the person with the higher level wins. The two players stop playing mid-game, and the player who is more skilled is the winner. The game's winner must then judge the skill of their opponent based on this relative level. In both cases, the probability of winning the next round is lower than the previous one.
At that time, Bayesian used the purchase of newly launched lottery tickets as an example to consider issues similar to the above. Suppose we don’t know the winning rate of the new lottery coupons, so we bought 10 randomly at the lottery store and ended up with 5 winning tickets. Then we estimate the probability of winning this lottery ticket is about 0.5.
However, suppose we only bought a lottery ticket and won the prize, then the probability of winning is 1?
This sounds unreasonable. Will the brokerage that issue lottery tickets be so dumb? Will you win every lottery ticket you buy? However, if the probability is not 1, what is the reasonable probability?
Later, the famous French astronomer and mathematician Pierre Simon, Marquis de Laplace, 1749–1827 provided a simple method to help us estimate the probability behind it.
Laplace published a paper titled "Treatise on the Probability of the Causes of Events" in 1774 to solve the above problems.
Suppose we repeat an experiment that will lead to success or failure, independently perform n times, and obtain s successes and n-s failures. What is the probability of success next time?
Laplace proposed a formula to predict the probability of success next time. This formula is:
P(next outcome is success)=(s+1)/(n+2)
This formula is called "Laplace’s Rule of Succession". Among them, n is the number of trials that occurred in the past (for example, 10 trials have been performed), s is the number of successes (for example, 5 successes), and the probability of success next time is (s+1)/(n+2).
For example, suppose that you bought 10 lottery tickets (n=10) and 5 of them won the prize (s=5). The probability of winning the next prize is 6/12, which is 0.5. At least the probability of winning does not go down. If there were 10 trains (n=10) on time in the past and 2 trains were delayed (s=2), the probability of the next train being delayed is 3/12=0.25, which means that the on-time rate of the next train has increased slightly.
Although Laplace's rule of continuation is simple and reasonable, it still drew some criticism. In the example in Wikipedia, according to the formula provided by Laplace, the probability of the sun rising tomorrow is:
P(sun will rise tomorrow)=(d+1)/(d+2)
Where d is the number of times the sun has risen in the past. Some people think this calculation is quite absurd, but Laplace believes that because the number of sun rises is too large (calculated in 4.5 billion years, the number of sun rises is 1.64 trillion), it can be inferred that the sun will still be the same tomorrow. It will still rise, and as long as it doesn't rain, you can still see the big sun appear in the east tomorrow morning.
Laplace's law of continuation provides a simple method for us to estimate the probability of success next time when faced with small data.