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what does bayes theorem tell us
A test catches 90 percent of sick people, and wrongly flags 10 percent of healthy ones. One person in a hundred is sick. You test positive. Sick, how likely?
Count real people, 1000 of them. Ten are sick, and the test catches nine. Of 990 healthy people, it wrongly flags 99. Everyone's in the tree.
Now look only at the positives: 9 sick, 99 healthy. So 9 out of 108 positives are truly sick, about 8 percent. Not 90.
That counting IS Bayes' theorem. The chance of sick given positive equals the chance of positive given sick, times the chance of sick, divided by the chance of any positive.
The prior, how common the thing was, carries huge weight. If 10 percent were sick, a positive means 50 percent. If half were sick, 90 percent.
So Bayes tells us how to update: begin with the prior, weigh how well the evidence fits, and land on a new belief. Thomas Bayes's essay, read in 1763, after his death, began it.
Bayes' theorem is the rule for letting evidence move your belief by exactly as much as it deserves, no more.