Bayesian updating with likelihood ratios

Use when

• New evidence arrives and you must revise a belief rather than replace it.
• A test, signal or indicator has a known or estimable error rate.
• People are treating a positive result as proof.

Do not use when

• There is no defensible base rate, which makes the posterior arbitrary.
• The evidence is not independent of what you already counted — you will double-count it.

Inputs required

• A prior, as odds
• How likely the evidence is if true, and if false

Procedure

1. 01

   State the prior as odds

   Odds, not percentages. A 5% base rate is 1:19 against. Odds make the update a multiplication instead of an equation.

2. 02

   Compute the likelihood ratio

   How likely is this evidence if the hypothesis is true, divided by how likely if false. A test with 90% sensitivity and 90% specificity gives 0.9 / 0.1 = 9.

3. 03

   Multiply

   Posterior odds = prior odds × likelihood ratio. 1:19 × 9 becomes 9:19.

4. 04

   Convert back if you must

   9:19 is about 32%. Convert only at the end — updating in percentages is where the arithmetic goes wrong.

5. 05

   Ask what would move it back

   Name the evidence that would push the odds the other way, and by how much. If nothing would, you are not reasoning from evidence.