Comment by godelski
5 days ago
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.
Again, my only argument was that you can have both situations occur. We could still construct a f∘g∘h to increase probability if we want. I'm not saying it cannot go down, I'm saying there's no absolute rule you can follow.
I don't think you can make a chain of logic f∘g∘h where the probability of the combined function is higher than the probability of f or g or h on their own.
Chain of logic meaning that only the last function updates the probability you care about, and the preceeding functions give you intermediate information that is only useful to feed into the next function.
It is an absolute rule you can follow, as long as you're applying it the way it was intended, to a specific organization of functions. It's not any kind of combining, it's A->B->C->D combining. As opposed to multiple pieces that each independently imply D.
Just because you can use ∘ in both situations doesn't make them the same. Whether x∘y∘z is chaining depends on what x and y and z do. If all of them update the same probability, that's not chaining. If removing any of them would leave you with no information about your target probability, then it's chaining.
TL;DR: ∘ doesn't tell you if something is a chain, you're conflating chains with non-chains, the rule is useful when it comes to chains
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