| H | The hypothesis: the thing you're unsure about, e.g. "it rains today". |
| E | The evidence: the news you observe, e.g. "the morning is overcast". |
| P(H) | The prior: your probability before seeing the evidence (Latin prior, "earlier"). Usually a base rate, such as "it rains on 20% of days this month". |
| P(E | H) | A likelihood: how often you'd see this evidence if H were true. The bar "|" is read "given". Example: on rainy days, mornings are overcast 70% of the time. |
| P(E | not H) | How often you'd see the same evidence if H were false. Example: on dry days, 20% of mornings are overcast. |
| P(H | E) | The posterior: your probability after the evidence (Latin posterior, "coming after"). This is what you actually want. Today's posterior becomes tomorrow's prior when new news arrives. |
| odds = p ÷ (1 − p) | Chance for against chance against. 20% → 0.2 ÷ 0.8 = 0.25, said "1 to 4". |
| LR | Likelihood ratio = P(E | H) ÷ P(E | not H). How many times more common the evidence is when H is true: 0.7 ÷ 0.2 = 3.5. This is the strength of the news. |
Where the odds form comes from
Bayes' rule is P(H | E) = P(E | H) · P(H) ÷ P(E). Write the same rule for "not H", then divide one by the other. The awkward P(E) term cancels and you get posterior odds = prior odds × LR. Rain example: 0.25 × 3.5 = 0.875, and 0.875 ÷ 1.875 = 46.7%.
See it as 1,000 cases
The clearest way to understand a posterior is to count. Each dot below is one day. Once you see the evidence, you are standing on one of the solid dots, so throw away the faded ones. The posterior is the share of solid dots that are green.
The classic trap
P(E | H) is not P(H | E). Suppose a system catches 90% of fraudulent trades and wrongly flags 5% of good ones, and 1% of trades are fraud. A flagged trade feels 90% likely to be fraud, but: prior odds 1:99, LR = 0.90 ÷ 0.05 = 18, posterior odds = 18/99 ≈ 0.18, so p ≈ 15%. A low base rate swamps even strong evidence. Set the sliders to 1%, 90%, 5% to see it.