Bayesupdating your beliefs
Belief before, evidence, belief after — the spam-filter arithmetic of changing your mind.
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An email arrives; the subject line says prize. Is this spam? Two true facts pull in opposite directions. Most spam does say prize — one point for spam. But most email is not spam in the first place — one point against. Both facts are true. The only way to weigh them against each other is to count.
So count. Take one thousand ordinary emails. In a batch like this, one hundred are spam, and the other nine hundred are normal. Now place the word. Prize appears in sixty of the one hundred spam emails, and in twenty of the nine hundred normal ones. Four counts, and the whole answer hides inside them.
The email in your hand says prize, so throw out every email that does not. Only eighty survive: sixty spam, twenty normal. Inside this smaller world, spam is sixty out of eighty — seventy-five percent. That is a reversal. You knew prize given spam: sixty percent of spam uses the word. You wanted spam given prize: seventy-five percent of prize emails are spam. Flipping the first into the second is Bayes' rule.
Why seventy-five? That count leaned on how much spam existed before you read a word: one hundred in every thousand. The starting amount is the prior — your belief before the evidence arrives. Now suppose only ten in a thousand were spam. Sixty percent of ten is six, so the prize pile holds six spam beside the same twenty normal. Six out of twenty-six: roughly twenty-three percent. Same evidence, weaker prior, much smaller answer. Rare things stay fairly rare, even after striking evidence.
That is all Bayes' rule is: counting, done after evidence shrinks the world. You start with a belief — the prior. Evidence arrives and throws out every case that does not match. You count what survives, and that count is your new belief — the posterior. One word moved an email from ten percent spam to seventy-five, and nothing was guessed. That is the arithmetic of changing your mind.