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Originally Posted by Matt R.
By the way, if you guys honestly want to know what Bayesian inference error you are making, you're assuming P(Amanda=guilty|evidence) is the same as P(evidence|Amanda=guilty).
Eh? This is your big revelation? It's not even coherent.
The probability that Amanda is guilty given the evidence is maybe 99%? This is the only thing that matters.
The probability that we have the evidence we do given that Amanda is guilty is very hard to quantify, but is maybe 20-80% ticking things off broadly. More specifically, given that there is a wide range of possibilities of what the evidence would look like in Amanda guilt, any
particular set of evidence given her guilt is actually quite improbable, because the set of possible evidence is so wide.
It's actually 239 who's making your error, so thanks for that. He has stated his confusion on this issue quite clearly. For example, he says there's no chance that no one in the group would snitch. He says that there's very small chance a large knife gets used. He's stated that it's incredibly improbable that none of her DNA/footprints are in the murder room (it's actually very common). He's talked about how incredibly rare female participants are (they're not). In 239 world, if the evidence suggests someone was killed by a group including a female, it's already long odds that she was framed, and all the evidence should be considered in that light.
He's making exactly the error you're talking about. He's taking all of these probabilities and summing them up and coming up with a very very low value. As would be the case in nearly any case; there's always missing evidence and things that don't make sense (not to mention, P(one person murders another) is very low to begin with.
He's basically working out P(evidence|Amanda=guilty), finding it very improbable (as a
specific set of evidence in most murder cases is), and then conflating that with P(Amanda=guilty|evidence).
So thanks for pointing out 239's fallacies. Perhaps you can educate him? I'm sure he's already grateful for your participation in this thread and would doubly appreciate schooling in Bayesian inference.
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Look up Prosecutor's fallacy and base rate fallacy if you want to understand more about how wrong you are about
I'm really looking forward to your response to Oski on this topic.