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Gyrodiot comments on Does Evidence Have To Be Certain? - Less Wrong
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Consider P(E) = 1/3. We can consider three worlds, W1, W2 and W3, all with the same probability, with E being true in W3 only. Placing yourself in W3, you can evaluate the probability of H while updating P(E) = 1 (because you're placing yourself in the world where E is true with certainty.
In the same way, by placing yourself in W1 and W2, you evaluate H with P(E) = 0.
The thing is, you're "updating" on an hypothetical fact. You're not certain of being in W1, W2, or W3. So you're not actually updating, you're artificially considering a world where the probabilities are shifted to 0 or 1, and weighting the outcomes by the probabilities of that world happening.
When you update, you're not simply imagining what you would believe in a world where E was true, you're changing your actual beliefs about this world. The point of updates is to change your behavior in response to evidence. I'm not going to change my behavior in this world simply because I'm imagining what I would believe in a hypothetical world where E is definitely true. I'm going to change my behavior because observation has led me to change the credence I attach to E being true in this world.
There's a labeling problem here. E is an event. The extra information you're updating on, the evidence, the thing that you are certain of, is not "E is true". It's "E has probability p". You can't actually update until you know the probability of E.
What the joint probability give you is by how much you have to update your credence in H, given E. Without P(E), you can't actually update.
P(H|E) tells you "OK, if E is certain, my new probability for H is P(H|E)". P(H|~E) tells you "OK, if E is impossible, my new probability for H is P(H|~E)". In the case of P(E) = 0.5, I will update by taking the mean of both.
Updating, proper updating, will only happen when you are certain of the probability of E (this is different form "being certain of E"), and the formulas will tell you by how much. Your joint probabilities are information themselves: they tell you how E relates to H. But you can't update on H until you know evidence about E.