So what ends up mattering is the ratio p[Tk | e+] : p[Tk | e-] I’m claiming that this ratio is likely to vary with k.
Wait, shouldn’t it be the ratio p[Tk & e+] : p[Tk & e-]? Maybe both ratios work fine for our purposes, but I certainly find it more natural to think in terms of &.
Unless I’ve confused myself badly (always possible!), I think either’s fine here. The | version just takes out a factor that’ll be common to all hypotheses: [p(e+) / p(e-)]. (since p(Tk & e+) ≡ p(Tk | e+) * p(e+))
Since we’ll renormalise, common factors don’t matter. Using the | version felt right to me at the time, but whatever allows clearer thinking is the way forward.
I’m probably being just mathematically confused myself; at any rate, I’ll proceed with the p[Tk & e+] : p[Tk & e-] version since that comes more naturally to me. (I think of it like: Your credence in Tk is split between two buckets, the Tk&e+ and Tk&e- bucket, and then when you update you rule out the e- bucket. So what matters is the ratio between the buckets; if it’s relatively high (compared to the ratio for other Tx’s) your credence in Tk goes up, if it’s relatively low it goes down.
Anyhow, I totally agree that this ratio matters and that it varies with k. In particular here’s how I think it should vary for most readers of my post:
for k>12, the ratio should be low, like 0.1.
for low k, the ratio should be higher.
for middling k, say 6<k<13, the ratio should be in between.
Thus, the update should actually shift probability mass disproportionately to the lower k hypotheses.
I realize we are sort of arguing in circles now. I feel like we are making progress though. Also, separately, want to hop on a call with me sometime to sort this out? I’ve got some more arguments to show you...
Wait, shouldn’t it be the ratio p[Tk & e+] : p[Tk & e-]? Maybe both ratios work fine for our purposes, but I certainly find it more natural to think in terms of &.
Unless I’ve confused myself badly (always possible!), I think either’s fine here. The | version just takes out a factor that’ll be common to all hypotheses: [p(e+) / p(e-)]. (since p(Tk & e+) ≡ p(Tk | e+) * p(e+))
Since we’ll renormalise, common factors don’t matter. Using the | version felt right to me at the time, but whatever allows clearer thinking is the way forward.
I’m probably being just mathematically confused myself; at any rate, I’ll proceed with the p[Tk & e+] : p[Tk & e-] version since that comes more naturally to me. (I think of it like: Your credence in Tk is split between two buckets, the Tk&e+ and Tk&e- bucket, and then when you update you rule out the e- bucket. So what matters is the ratio between the buckets; if it’s relatively high (compared to the ratio for other Tx’s) your credence in Tk goes up, if it’s relatively low it goes down.
Anyhow, I totally agree that this ratio matters and that it varies with k. In particular here’s how I think it should vary for most readers of my post:
for k>12, the ratio should be low, like 0.1.
for low k, the ratio should be higher.
for middling k, say 6<k<13, the ratio should be in between.
Thus, the update should actually shift probability mass disproportionately to the lower k hypotheses.
I realize we are sort of arguing in circles now. I feel like we are making progress though. Also, separately, want to hop on a call with me sometime to sort this out? I’ve got some more arguments to show you...