This is a very enlightening post. But something doesn’t seem right. How can receiving a random bit cause Beauty to update her probability, as in the case where Beauty is an AI? If Beauty already knows that she will update her probability no matter what bit she receives, then shouldn’t she already update her probability before receiving the bit?
That’s a good point, but let’s consider where that principle comes from: it derives from the fact that
Pr(A∣M)=Pr((A & B1) or … or (A & Bn)∣M)=∑iPr(A & Bi∣M)=∑iPr(A∣Bi,M)Pr(Bi∣M)
where B1,…,Bn are mutually exclusive and exhaustive propositions. The second equality above relies on the fact that the Bi are MEE; otherwise we’d have to subtract a bunch of terms for various conjunctions (ANDS) of the Bi. But the set of propositions
X2(y)≜R(y,M) or R(y,T),
indexed by y, are not mutually exclusive. If y and y′ are remembered perceptions on different days, then bothX2(y) and X2(y′) will be true.
What I wrote above may be a bit misleading. The issue isn’t that you have additional terms for conjunctions of the Bi, but that the weights Pr(Bi∣M) sum to more than 1. In particular, consider the case when AI Beauty gets exactly one bit of input. Then for y=0 or 1,
And yet if you just keep them split up as R(y,d) indexed by both y and d, the MEE condition holds. So if Beauty expected to get both the observations and be told the day of those observations, she would expect no net update of P(H).
Huh. Does this mean that if being told only the content y makes an agent predictably update towards P(H)<0.5, being told only the day d makes your procedure predictably update towards P(H)>0.5?
This is a very enlightening post. But something doesn’t seem right. How can receiving a random bit cause Beauty to update her probability, as in the case where Beauty is an AI? If Beauty already knows that she will update her probability no matter what bit she receives, then shouldn’t she already update her probability before receiving the bit?
That’s a good point, but let’s consider where that principle comes from: it derives from the fact that
where B1,…,Bn are mutually exclusive and exhaustive propositions. The second equality above relies on the fact that the Bi are MEE; otherwise we’d have to subtract a bunch of terms for various conjunctions (ANDS) of the Bi. But the set of propositions
indexed by y, are not mutually exclusive. If y and y′ are remembered perceptions on different days, then both X2(y) and X2(y′) will be true.
What I wrote above may be a bit misleading. The issue isn’t that you have additional terms for conjunctions of the Bi, but that the weights Pr(Bi∣M) sum to more than 1. In particular, consider the case when AI Beauty gets exactly one bit of input. Then for y=0 or 1,
and 5/8+5/8=5/4>1. If we try the same decomposition as in my previous comment, then usingPr(H∣X2(y),M)=1/2.5=2/5, we find
and everything is still consistent.
And yet if you just keep them split up as R(y,d) indexed by both y and d, the MEE condition holds. So if Beauty expected to get both the observations and be told the day of those observations, she would expect no net update of P(H).
Huh. Does this mean that if being told only the content y makes an agent predictably update towards P(H)<0.5, being told only the day d makes your procedure predictably update towards P(H)>0.5?