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  1. Pr(Sx ∣ x∈𝓗) ≫ 0
  2. Pr(Sx ∣ Tx & x∈𝓗) ≫ 0
  3. Pr(Sx ∣ ¬Tx & x∈𝓗) ≪ 1

Therefore:

  1. Pr(Tx ∣ Sx & x∈𝓗) ≫ 0

Is this an instance of the base-rate fallacy, or is this line of reasoning valid? It seems to me that (4) follows from (1)-(3), but would not follow from (2)-(3) alone (indeed, that would be an instance of the base-rate fallacy). Am I right here?

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You're right. Here is a counter-example. Suppose T means having a very rare disease : among 1000 persons only 10 have it.

Suppose S means having blond hair.

(2) says that most of the 10 persons having the disease have blond hair. Imagine it's 9 persons.

(3) says few of the 990 persons without the disease have blond hair. Imagine it's 99.

(4) says most people with blond hair have the disease, which in this case is false: 99 don't have it, only 9 have it.

Now add clause (1): most people have blond hair. Then the disease cannot be a very rare disease after all (that would be inconsistent with (3)) and (4) turns out true:

(1) Most people have blond hair. Imagine 900 have blond fair, 100 don't.

(3) most people without the disease are among the 100 non-blond. People without the disease cannot exceed 100 by large. Imagine it's 110 and only 20 are blond

(2) most people with the disease have blond hair. This is already constrained by (1) and (3): 880 over 890.

From which (4) follows as well: most people with blond hair have the disease: here 880 over 900. Obviously no counterexample will be found in this case because we started with no particular assumption.

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