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Keep the event and evidence in the notation
| Term | Quantity | Question |
|---|---|---|
| Prior | P(A) | How much probability does the model assign to A before E? |
| Likelihood | P(E|A) | How likely is E if A holds? |
| Posterior | P(A|E) | How much probability belongs to A after conditioning on E? |
Include the evidence under the complement
In a binary model, the evidence can arise under A or under not A. The total probability P(E) therefore includes both branches. Ignoring the complement branch can falsely imply that every observed E must have come from A.
Normalize the event-and-evidence branch
Bayes’ rule divides P(A∩E) by the total evidence probability, provided P(E) is positive. The likelihood is only one part of that calculation; it is not generally the posterior.
Both branches contribute to the denominator in this two-hypothesis model.
Check evidence that does not distinguish the branches
If E has the same positive likelihood under A and not A, the common likelihood cancels and the posterior equals the prior. Evidence only changes this model’s event probability when its relative likelihood differs between the branches.