Laplace's rule of succession. Laplace claimed that when an...

Laplace's rule of succession. Laplace claimed that when an event happens $n$ times in a row and never fails to happen, the probability that the event will occur the next time is $(n+1) /(n+2) .$ Can you suggest a rationale for this claim?

Key Concepts

Posterior Predictive Probability

Posterior predictive probability is the probability of a future event predicted by averaging over the posterior distribution of a parameter. Under the assumption of a uniform prior and after observing n successes, the posterior predictive probability that the next trial will be a success is calculated by integrating over the updated Beta distribution, yielding the result (n+1)/(n+2). This provides the rationale for Laplace's rule of succession.

Beta Distribution

The Beta distribution is a family of continuous probability distributions defined on the interval between 0 and 1 and is the conjugate prior for the binomial distribution in Bayesian analysis. When a uniform prior (a Beta distribution with parameters 1 and 1) is used, and n successes are observed, the resulting posterior distribution is a Beta distribution with parameters (n+1) and 1. This conjugacy facilitates the analytical derivation of the rule of succession.

Bayesian Inference

Bayesian inference is a statistical framework that updates the probability of a hypothesis as more evidence becomes available. It combines prior beliefs (expressed through a prior distribution) with observed data (through the likelihood) to produce updated beliefs (the posterior distribution). This framework is key to understanding Laplace's rule of succession because it explains how observed successes can update our belief about the likelihood of future outcomes.

Prior Distribution

A prior distribution encapsulates our beliefs about the unknown parameter before any data are observed. In the context of Laplace's rule, a uniform prior is typically assumed, indicating no initial preference for any particular value of the probability. This noninformative prior is essential because it ensures that the data drive the updating process, leading to the formula (n+1)/(n+2) after observing n successes.

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