Bayes' Theorem

The agency buys a fake-listing detector. It catches 90% of fake listings and wrongly flags only 10% of genuine ones. A listing gets flagged. Is it 90% likely to be fake? Most people say yes, and most people are badly wrong. Bayes' theorem shows why.

The rule

Reverse a condition: likelihood times prior, divided by evidence.

The detector tells you $P(\text{flag} \mid \text{fake})$. You want the reverse, $P(\text{fake} \mid \text{flag})$. Bayes' theorem connects them:

• Prior $P(A)$: how common the cause is before any evidence. Here 5% of listings are fake.
• Likelihood $P(B \mid A)$: how often the evidence appears when the cause is true. Here 0.9.
• Evidence $P(B)$: how often the evidence appears at all, from any cause.
• Posterior $P(A \mid B)$: the updated belief after seeing the evidence.

Check it on last lesson's 20 flats: $P(\text{near} \mid \text{fast}) = 0.75 \times 0.4 / 0.6 = 0.5$, the same answer the table gave by counting.

Bayes' theorem also chains. If a second, independent check flags the same listing, today's posterior of 0.32 becomes the prior for that update, and the belief climbs again. Learning from evidence is repeated Bayesian updating.

A naive Bayes classifier uses this rule directly: it multiplies the likelihoods of each feature (each word in a listing, say) with the prior for each class, and picks the class with the higher result.

The shortcut

Natural frequencies. Imagine a round number of cases, such as 1000, and split them down a tree. Then the posterior is a simple fraction you can read off, with no formula to misremember.

The detector is good, but fakes are rare, so a small false-alarm rate applied to the large genuine group produces more flags than the real fakes do. Ignoring the prior is called base-rate neglect, and it is the trap this lesson is about.

Drills

Answer fast — trust the rule.

Remember

• Reverse a condition: likelihood times prior, divided by evidence.
• Shortcut: imagine 1000 cases, split them down a tree, read off the fraction.
• Trap: base-rate neglect, forgetting how rare the cause is.