Now that I write it out explicity, I see that, while it isn’t circular, it’s definitely double-counting. I’m not sure that’s a problem, though. Initially, for all deterministic programs q that model the environment, it calculates its expected reward assuming each q one at a time. Then it weights all q by the rewards and acts to maximize the expected reward for that weighted combination of all q.
Oh, I thought the point was that in universes where the goldpan is true, the universe is determined by the set of things that would give you high utility if you believed them. Which doesn’t sound coherent to me (for one thing, there are probably multiple sets of such beliefs), so maybe you understood the meaning better than I...
It looks to me like the goldpan is circular, you need a probability to calculate expected utility to calculate prior probability.
Consider this ugly ASCII version of the expression for AIXI found in this paper by Marcus Hutter,
a_k := arg max[a_k, SUM[o_k*r_k … max[a_m, SUM[o_m*r_m, (r_k +...+ r_m) SUM[q:U(q,a_1...a_m) = o_1*r_1..o_m*r_m, 2^-l(q)] ]]...]] .
What I was thinking was to replace the inner sum for the Solomonoff prior, SUM[q:..., 2^-l(q)], with a repeat of the interleaved maxes and SUMs.
SUM[q:U(q,a_1...a_m)=o_1*r_1..o_m*r_m, max[a_k, SUM[o_k*r_k … max[a_m, SUM[o_m*r_m, (r_k + … + r_m)]]...]] ] .
Now that I write it out explicity, I see that, while it isn’t circular, it’s definitely double-counting. I’m not sure that’s a problem, though. Initially, for all deterministic programs q that model the environment, it calculates its expected reward assuming each q one at a time. Then it weights all q by the rewards and acts to maximize the expected reward for that weighted combination of all q.
“I believe that I will get nigh-unbounded utility after I die, but I can’t cause my death. ”
Oh, I thought the point was that in universes where the goldpan is true, the universe is determined by the set of things that would give you high utility if you believed them. Which doesn’t sound coherent to me (for one thing, there are probably multiple sets of such beliefs), so maybe you understood the meaning better than I...