All numeric cases below are synthetic teaching examples, not empirical research findings.
Example: replicated announcement
| Prior on-time launch probability is 0.30. P(announcement | on time)=0.8; P(announcement | late)=0.2. |
Posterior=0.24/(0.24+0.14)=0.6315789. Two outlets repeating the same announcement do not multiply the LR twice.
Example: weak signal
Prior=0.4, a=0.6 and b=0.5 give posterior=0.24/(0.24+0.30)=0.444444. A vivid observation can warrant a small update.
Counterexample
“The source is 90% reliable, so multiply prior odds by 9” is not generally valid. What reliability means, base rates and the process generating true/false claims must be specified.
Model failure
Prior=0, a=1, b=0 gives a zero denominator. The helper must reject this impossible-under-model observation.