Module IV of VI

Weighing evidence

A guest lies dead in the library of a country house. Eight people were inside when it happened, the butler among them, and you — the detective — hold no evidence yet. Honest starting odds on the butler: 1 : 7. One chance he did it, seven that someone else did.

Then the clues start arriving, and each one should move you by a measurable amount. How much? Ask of every clue exactly one question: how much more expected is this if he’s guilty than if he’s innocent? Fingerprints on the cellar key are quite likely if the butler did it, and fairly unlikely (though possible — he does work here) if not; say eight times more expected. That multiplier is the clue’s likelihood ratio, and it is the entire measure of evidential strength. Cat hair on his jacket is almost equally expected either way — everyone pets the cat — so its ratio sits near 1, and a ratio of 1 is a polite way of saying no evidence at all.

Panel IV·1 — The case against the butler

prior 1 : 7 · toggle the clues

Eight people were in the house, so the butler starts at odds 1 : 7. Turn clues on and off; each multiplies the odds by its strength.

how the odds stand

13 %

1 : 7 = 1 : 7 → P(butler | clues) = 13 % · not even the likeliest suspect yet

Try this Turn on the fingerprints alone, then the cat hair alone, and compare the damage. Then stack every clue except the fingerprints — can four weak clues outvote one strong one?

What the panel is doing is one multiplication, repeated:

posterior odds = prior odds × likelihood ratio

That’s Bayes’ theorem in its working clothes. Each clue multiplies; three clues of strength 8, 4, and 2 together multiply the odds by 64. Weak evidence stacks — four ratios of 1.5 multiply to about 5 — which is why a case can be built from circumstance, and also why a jury should ask whether the “independent” clues really are independent. (If the cook who breaks the alibi also found the fingerprints, those are not two clues. They’re one clue wearing two hats.)

You have now computed P(H | E) — the probability of a hypothesis given evidence — several times, so the notation can finally be introduced as a label for something you already own. Written out in full:

P(H | E) = P(E | H) · P(H) / P(E)

Read it as bookkeeping, with each piece something you’ve already touched: P(H) is the prior (Module III), P(E | H) is how expected the evidence is under the hypothesis — the top half of a likelihood ratio — and the division by P(E) just rescales so the probabilities sum to one. The dots-in-a-grid counting of Module II is this formula; the odds form above is the same statement rearranged for mental arithmetic.

The notation also names the classic courtroom disaster. P(E | H) and P(H | E) are different numbers — “this DNA match would occur in one person in a million” is a statement about evidence given innocence, and it is not the probability of innocence given the evidence, no matter how confidently a prosecutor reads it aloud. Swapping them is called the prosecutor’s fallacy, it has helped convict innocent people, and you did the correct version by counting dots two modules ago.

Retell itJudge every clue by one question — how surprised would I be to see this if I were wrong? — and multiply.

So far all our beliefs have been about yes-or-no questions. Module V lets the belief itself grow a shape.