Tools
Bayes Primer
Start with a prior belief, observe evidence, and let Bayes' rule update the belief. In the medical screening scenario, seeing the evidence moves the chance of H from 1.00% to 16.67%.
Scenario
A rare condition with a good test can still produce many false alarms.
Ten thousand cases
Picture 10,000 cases. Some fraction have H, the thing you care about. The evidence shows up often among those cases and, if it is any good, rarely among the rest. After you see it, only the cases that show it are still in play.
Chance a randomly selected person truly has the condition.
If a person has the condition, chance the test reports positive.
If a person does not have the condition, chance the test still reports positive.
The top bar splits the cases by H; the middle keeps only those that show the evidence. The 1 false negative and 9,405 true negatives drop out. The bottom bar stretches the survivors back to full width, and its dark share is the posterior: about 1 in 6.0 positives actually has H.
The rule
The bottom bar is Bayes' rule drawn to scale. The posterior is the true positives divided by everyone who shows the evidence:
The denominator counts both kinds of positive, those with H and those without:
Here that comes to 5.94%. A rare H makes the second term large even when the false positive rate is small, which is why a good test for a rare condition still raises many false alarms.
How much the prior matters
Hold the evidence fixed and vary only the prior. The same evidence that barely moves a rare hypothesis can make a common one nearly certain.
Posterior against prior for this evidence, which is 20 times as likely with H as without. The dashed diagonal is no update at all; the vertical segment is how far the evidence moves the current prior of 1%. The prior axis is logarithmic so that rare hypotheses, where the base rate matters most, get room.