4 ms·

Yeah, now that I think more about it, I think I was confused myself. Specifically, I think I got the math right but the example wrong (and I confused you too).

Yeah, now that I think more about it, I think I was confused myself. Specifically, I think I got the math right but the example wrong (and I confused you too). Let me try again. I think that Deutsch is saying that h is the proposition "smoker implies cancer", and e is a specific instance of a person where the hypothesis holds (either a nonsmoker or a smoker with cancer). He is talking about e being instances of h, so h must be a higher order proposition about instances. But now h can be decomposed as we said into a logically necessary part (h|e) and a part (h|~e) that may be true or false depending upon which universe you live in. By the argument above, finding more instances of the theory should decrease our belief in the (h|~e) part. Since h|~e is the same as e->h, gathering more e should decrease our faith that the evidence validates the hypothesis. Presumably Deutsch is saying that the logically necessary part is sort of trivial (a mere theorem) whereas (h|~e) has actual physical content, so why do we believe that the evidence increases our confidence in the physical portion of the hypothesis? By the way, I got all this math from the unapproachable paper, which is not that unapproachable if one looks at the math alone. Like you, I am trying to figure out how this math applies to the real world.

justinpombrio 4y ago
So if we observe e, then: - h|e becomes 1 - h|~e = e->h decreases - h increases I notice I'm confused, though. Is h|e about this specific instance of e, or the general e? How about e->h? It feels like we're moving the goal-posts: at first h|e = 1 because we're talking about a specific e (we found a smoker with cancer), and e->h decreases because we're talking about that specific e, but then later we start to draw conclusions about the general e->h such as describing it as whether "the evidence validates the hypothesis". And I don't know how to formally relate the specific e->h to the general e->h.
justinpombrio 4y ago
> Specifically, I think I got the math right but the example wrong (and I confused you too). Don't worry, I was much more confused before you came along :-). I also followed the math in the paper but didn't know what devastating contradiction for Bayesianism it was oh-so-darkly hinting at. I keep trying to describe what happens if e and h are both general, and bouncing off. Let's try a different assumption: e is the statement "the first person tested is consistent with smoking->cancer". And in fact, we just tested them, and they're a smoker and have cancer. So e is unambiguously true. And h is the hypothesis "smoking -> cancer". Then: - e is 1 - h has increased - h|e has increased, in fact it is 1 - h|~e has decreased h|e is tautologically true, once we have observed e. And h|~e is e->h, the amount to which this observation implies "smoking->cancer". Which seems odd, though I don't have an immediate intuition about how an observation is supposed to effect, not the hypothesis that it supports, but the implication between itself and that hypothesis. But maybe that's the darkly-hinted-at problem?

justinpombrio 4y ago

So if we observe e, then: - h|e becomes 1 - h|~e = e->h decreases - h increases I notice I'm confused, though. Is h|e about this specific instance of e, or the general e? How about e->h? It feels like we're moving the goal-posts: at first h|e = 1 because we're talking about a specific e (we found a smoker with cancer), and e->h decreases because we're talking about that specific e, but then later we start to draw conclusions about the general e->h such as describing it as whether "the evidence validates the hypothesis". And I don't know how to formally relate the specific e->h to the general e->h.

justinpombrio 4y ago

> Specifically, I think I got the math right but the example wrong (and I confused you too). Don't worry, I was much more confused before you came along :-). I also followed the math in the paper but didn't know what devastating contradiction for Bayesianism it was oh-so-darkly hinting at. I keep trying to describe what happens if e and h are both general, and bouncing off. Let's try a different assumption: e is the statement "the first person tested is consistent with smoking->cancer". And in fact, we just tested them, and they're a smoker and have cancer. So e is unambiguously true. And h is the hypothesis "smoking -> cancer". Then: - e is 1 - h has increased - h|e has increased, in fact it is 1 - h|~e has decreased h|e is tautologically true, once we have observed e. And h|~e is e->h, the amount to which this observation implies "smoking->cancer". Which seems odd, though I don't have an immediate intuition about how an observation is supposed to effect, not the hypothesis that it supports, but the implication between itself and that hypothesis. But maybe that's the darkly-hinted-at problem?