Core identities
For prior p and likelihood ratio LR=P(E|H)/P(E|not H): prior odds=p/(1-p); posterior odds=prior odds*LR; posterior probability=posterior odds/(1+posterior odds).
An equivalent and numerically useful expression is: p_new=(pa)/(pa+(1-p)*b), where a=P(E|H) and b=P(E|not H).
If the denominator is zero, E is impossible under the specified model. Report model failure, not 0%, 50% or a silent epsilon replacement. Dogmatic priors 0 or 1 cannot generally learn from finite likelihood evidence; avoid them for ordinary uncertain events.
Likelihood elicitation
| Ask “How often would this exact evidence appear if H were true?” and separately “How often if H were false?” This is not the inverse question P(H | E). Source reliability is not itself a likelihood ratio. |
Distinguish evidence that is common under both hypotheses from evidence that discriminates. A polished demo could be likely both for an on-time product and for a late product.
When numeric likelihoods are not measured, label them as judgmental assumptions. Produce sensitivity to plausible likelihood pairs or state that only a qualitative update is justified. Do not back-solve a likelihood ratio simply to launder a preferred final probability.
Evidence dependence
For E1 then E2, the second multiplier is: P(E2|H,E1)/P(E2|not H,E1).
Two outlets copying one press release are one evidence origin. If E2 deterministically repeats E1, the conditional LR is 1. Separate genuinely new observations from wider publicity of an old observation. Also check whether a “new” test was already part of the base-rate class or prior estimate.
Absence is evidence only when an observation process makes absence diagnostic. Not seeing an incident in an incomplete feed does not establish no incident. Model detection and publication likelihoods when relevant.
Robustness and limits
Run the low, central and high defensible likelihood settings. Keep sensitivity ranges distinct from statistical credible intervals. A likelihood-ratio calculation is internally valid conditional on its inputs; that alone does not validate the inputs.
Use current, task-authorized evidence for live forecasts. For historical cutoffs, restrict sources to that cutoff and disclose possible pretraining contamination. Do not present a retrospective exercise as prospective model validation.
Helper
Run Python 3 with the path resolved relative to this skill directory:
python3 scripts/bayes.py --prior 0.30 --p-e-h 0.8 --p-e-not-h 0.2
The script is offline and uses only the standard library. It returns a calculation, not an evidence-quality judgment. If execution is unavailable, compute the identity explicitly and disclose that no script ran.