Bayesian Method: Updating Beliefs

Bayes' Theorem: Formula for Rational Updating

Thomas Bayes (18th century) formulated a theorem that became the foundation of Bayesian statistics and rational thinking: P(H|E) = P(E|H) × P(H) / P(E).

P(H) — the "prior" probability of the hypothesis before obtaining evidence. P(E|H) — the probability of observing evidence E, given that hypothesis H is true (likelihood). P(H|E) — the "posterior" probability of the hypothesis after obtaining evidence. P(E) — the probability of the evidence (normalizing factor).

This is the formula for rational updating of beliefs. Bayesian thinking: you have a "prior" confidence in a hypothesis. You observe new evidence. Posterior confidence is a function of your prior confidence and the strength of the evidence.

Bayesian Thinking in Practice

Applications of the Bayesian method have spread far beyond academic statistics. Spam filters (Bayesian classifier): the probability that an email is spam is updated with every new word.

Medical diagnostics: based on symptoms, we update the probabilities of different diagnoses. Machine learning (naive Bayesian classifier). Search for submarines and airplanes (the MH370 search used Bayesian methods to update search areas).

"Superforecasters" (Tetlock, "Superforecasting"): people who systematically make more accurate forecasts are, as a rule, Bayesian thinkers: they start with a base rate, update when new data arrives, avoid "wishful thinking", and are ready to change their opinion.

The opposite: "confirmation bias" — we seek evidence that confirms beliefs already held and ignore contradictory information. This is not Bayesian updating; it is a Bayesian error.

Question for reflection: When was the last time you significantly changed your opinion based on new evidence? Are there professional beliefs you "do not update", despite receiving contradictory data?