Laplaceszabály

Laplaceszabály, also known as Laplace's rule of succession, is a concept in probability theory that deals with estimating the probability of a future event given past observations. It is particularly useful when dealing with events that have not yet occurred and for which there is no prior knowledge about their probabilities.

The rule states that if an event has occurred $s$ times in $n$ independent trials, the probability of the event occurring in the next trial is estimated as $(s+1)/(n+2)$. This formula is derived from a Bayesian approach where a uniform prior distribution is assumed for the probability of the event. The uniform prior represents a state of complete ignorance about the event's likelihood before any observations are made.

Laplace's rule is often applied in situations where we want to make predictions based on limited data. For example, if a coin has been flipped 10 times and landed heads 6 times, Laplace's rule would estimate the probability of getting heads on the next flip as (6+1)/(10+2) = 7/12. This is in contrast to the frequentist approach, which would estimate the probability as 6/10. The "+1" and "+2" in the numerator and denominator, respectively, represent the addition of two hypothetical "prior" observations, one success and one failure, which helps to smooth the probability estimate, especially when the number of trials is small.

The rule has been subject to some criticism, particularly regarding the assumption of a uniform prior, which may not always be appropriate. However, it remains a simple and intuitive method for estimating probabilities in the absence of strong prior information.