Bayesian Updating: Reasoning From Evidence

## The Bayesian Approach Bayesian reasoning is a formal framework for updating beliefs in response to new evidence. Rather than forming fixed opinions, Bayesian thinkers hold beliefs as probabilities and update those probabilities when new information arrives. ## The Basic Formula P(A|B) = P(B|A) × P(A) / P(B) Where: - P(A) = prior probability of A before new evidence - P(B|A) = probability of observing evidence B given A is true - P(B) = total probability of observing evidence B - P(A|B) = posterior probability of A after observing B In plain language: start with your prior belief. Update it based on how likely the new evidence would be if your belief were true. ## Betting Application **Prior:** Your model gives Team A a 50% win probability. **New information:** Key centre-back is confirmed absent. **Bayesian update:** - What is the historical win rate for Team A without this player? (-5% vs average) - Revised probability: 45% **Further update:** Heavy rain forecast. - Historical data: heavy rain reduces home win rate by 3% in this league - Revised probability: 42% The Bayesian approach gives a principled mechanism for incorporating new information without abandoning the prior estimate entirely. ## The Key Discipline Bayesian updating requires honest prior estimates (not anchored to market prices) and honest assessment of how diagnostic the new evidence is. "The coach said they're confident" is low-diagnostic evidence — it changes the probability by very little. "The starting goalkeeper is confirmed out" is high-diagnostic — it changes the probability significantly.