How should evidence move a belief?

A sheet for changing your mind on purpose. Start with a priorYour honest belief before looking at the evidence in front of you — usually anchored on the base rate, i.e. how often this kind of claim turns out true., then weigh each clue by how often you'd expect to see it if the claim were true versus if it weren't. The gap between those two worlds is the only thing that moves the needle — Bayes just folds it in, one clue at a time.

The claim
Before any evidence, the chance thatis%.
That's prior odds of about 1-to-5.7.
The evidence, one clue at a time

Read each row as a short sentence: how often you'd expect the clue in each world — if the claim is trueThe likelihood ratio is the only thing that moves a belief: how many times more (or less) likely you'd see this exact clue in a world where the claim holds than in one where it doesn't. Equal in both worlds means no pull at all. versus if it's false. Their ratio becomes the clue's pull (× supports, ÷ argues against). The gauge on the right tracks the running beliefAlso called the posterior: the chance the claim is true after folding in this clue and every clue above it. Each row updates the one before. after this clue, and the tick marks where it stood just before.

  1. 1
    I'd expect to see this% of the time if true,% if false.
    ×8 · supports58.5%The chance this alert is a real incident is true after folding in this clue: 58.5% (it was 15% just before). This carries forward as the starting point for the next clue.▲ 43.5 ptsClue 1 raised the belief by 43.5 points, from 15% to 58.5%.
  2. 2
    I'd expect to see this% of the time if true,% if false.
    ÷2 · argues against41.4%The chance this alert is a real incident is true after folding in this clue: 41.4% (it was 58.5% just before). This carries forward as the starting point for the next clue.▼ 17.2 ptsClue 2 lowered the belief by 17.2 points, from 58.5% to 41.4%.
  3. 3
    I'd expect to see this% of the time if true,% if false.
    ×12 · supports89.4%The chance this alert is a real incident is true after folding in this clue: 89.4% (it was 41.4% just before). This carries forward as the starting point for the next clue.▲ 48.1 ptsClue 3 raised the belief by 48.1 points, from 41.4% to 89.4%.
89.4%posterior beliefThe posterior: your updated probability after folding in every clue. Prior odds × every likelihood ratio, converted back to a probability.

After 3 clues, the chance that this alert is a real incident moves from 15% to 89.4% — a rise of 74.4%.

15%prior
89.4%posterior
+16.8 dBevidence weightThe total weight of all the evidence, in decibans — the log-odds shift from prior to posterior. Each clue's weight simply adds to it.
How the belief moved
0%25%50%75%100%Prior: 15%priorError rate spiked 10×: 58.5%1Other regions look healthy: 41.4%2Customers are reporting it: 89.4%315%89.4%

Each clue nudges the belief up (it fits this alert is a real incident) or down (it fits the alternative). Notice the curve is steepest through the middle and flattens near 0 % and 100 % — the same clue moves an uncertain belief far more than one that's already near-settled.

Weight of evidenceDecibans: 10·log₁₀ of the odds. Alan Turing's wartime unit for the weight of evidence — because in log-odds, independent clues just add up, the same trick that turns multiplying probabilities into stacking bars.
Prior · 15%
−7.5 dB
Error rate spiked 10×
+9.0 dB
Other regions look healthy
−3.0 dB
Customers are reporting it
+10.8 dB
Posterior · 89.4%
+9.3 dB

Left of centre argues against this alert is a real incident; right of centre supports it. Because the bars are log-odds, the prior and every clue literally stack end to end into the posterior — multiplying odds becomes adding weights. A clue pointing the wrong way doesn't veto the others; it just subtracts its length. That's why one healthy signal can't cancel two strong alarms.

Bayes in odds form: posterior odds = prior odds × every likelihood ratio