Comment by godelski
5 days ago
Definitely. Real world adds lots of complexities and nuances, but I was just trying to make the point that it matters how those inferences compound. That we can't just conclude that compounding inferences decreases likelihood
Well they were talking about a chain, A->B, B->C, C->D.
You're talking about multiple pieces of evidence for the same statement. Your tests don't depend on any of the previous tests also being right.
Be careful with your description there, are you sure it doesn't apply to the Bayesian example (which was... illustrative...? And not supposed to be every possible example?)? We calculated f(f(f(x))), so I wouldn't say that this "doesn't depend on the previous 'test'". Take your chain, we can represent it with h(g(f(x))) (or (f∘g∘h)(x)). That clearly fits your case for when f=g=h. Don't lose sight of the abstractions.
So in your example you can apply just one test result at a time, in any order. And the more pieces of evidence you apply, the stronger your argument gets.
f = "The test(s) say the patient is a vampire, with a .01 false positive rate."
f∘f∘f = "The test(s) say the patient is a vampire, with a .000001 false positive rate."
In the chain example f or g or h on its own is useless. Only f∘g∘h is relevant. And f∘g∘h is a lot weaker than f or g or h appears on its own.
This is what a logic chain looks like, adapted for vampirism to make it easier to compare:
f: "The test says situation 1 is true, with a 10% false positive rate."
g: "If situation 1 then situation 2 is true, with a 10% false positive rate."
h: "If situation 2 then the patient is a vampire, with a 10% false positive rate."
f∘g∘h = "The test says the patient is a vampire, with a 27% false positive rate."
So there are two key differences. One is the "if"s that make the false positives build up. The other is that only h tells you anything about vampires. f and g are mere setup, so they can only weaken h. At best f and g would have 100% reliability and h would be its original strength, 10% false positive. The false positive rate of h will never be decreased by adding more chain links, only increased. If you want a smaller false positive rate you need a separate piece of evidence. Like how your example has three similar but separate pieces of evidence.
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