The Beta Distribution for Probability Estimation

## Modelling Uncertainty in Probabilities You estimate that a team wins 52% of matches. But how confident are you in this estimate? The beta distribution models uncertainty about a probability itself. ## The Beta Distribution Parameters The beta distribution β(α, β) on [0,1] has: - Mean = α / (α + β) - Variance = αβ / ((α+β)² × (α+β+1)) **Interpretation:** You can think of α as the number of "successes" observed and β as the number of "failures." A team that has won 52 of their last 100 matches has α = 52, β = 48. ## Using Beta for Credible Intervals The 95% credible interval for team win probability from 100 matches (52 wins, 48 losses): Using the beta distribution: approximately [42%, 62%] This is the Bayesian equivalent of a confidence interval: "I am 95% confident the true win probability lies between 42% and 62%." ## Prior Information and Bayesian Updating The beta distribution naturally incorporates prior information: Prior: Team is average (β(50, 50) → 50% win probability with moderate confidence) New data: 10 matches, 6 wins Posterior = β(50+6, 50+4) = β(56, 54) → mean = 56/110 = 50.9% The prior tempers the new information: we do not leap from 50% to 60% after just 10 matches. As more matches accumulate, the posterior approaches the pure observed win rate — the data gradually dominates the prior.