Rule of succession - Wikipedia
bayesianprobabilitylaplaceinductive-reasoningpseudocounts
Abstraction: Laplace's formula estimating next-trial success probability from observed counts
Key points:
- Introduced by Laplace for the sunrise problem: given s successes in n trials, probability of next success is (s+1)/(n+2) — equivalent to adding one pseudocount per outcome
- Mathematical derivation: uniform prior on p, binomial likelihood, beta posterior Beta(s+1, n-s+1) with expected value (s+1)/(n+2)
- Laplace applied it to compute ~1,826,200:1 odds the sun will rise tomorrow (5000 years * 365.25 days), while acknowledging the prior is inappropriate given physical knowledge
- Generalizes to m categories via Dirichlet-multinomial: probability of next observation in category i is (n_i + 1)/(n + m), adding one pseudocount per category
- Rule only strictly valid when prior knowledge is exactly "m outcomes are possible but nothing more"; applying it in other contexts answers a different question
- Not valid when s=0 or s=n with total-ignorance improper prior (posterior becomes improper); rule of succession implicitly encodes "both success and failure are possible"
Connections: Pierre Simon Laplace · Bayesian Inference · Prior Probability · Inductive Reasoning