An idea I discussed with @Leon Lang that did not make it into the post is to look for a Friedberg numbering of the l.s.c. semimeasures, which could be used to control the weights P_ap. A Friedberg numbering for the recursively enumerable sets / partial recursive functions is (just) a unique effective numbering, which surprisingly enough does exist (though it is not admissable, meaning it can’t be translated to the standard numbering based on reasonable machine descriptions). If such a numbering existed, and U were chosen to implement it, then each l.s.c. semimeasure \nu would get prior weight (P_ap) exactly 2^(-l(p*)) where p* is the shortest program such that U(p*, _) generates the distribution of \nu.
l(p*) is one intuitive (but highly uncomputable) definition for K(\nu), but the normal definition is K(i) where i is the index of \nu (:= \nu_i) is some computable enumeration of l.s.c. semimeasures. Therefore, P_ap would still not match 2^-K(\nu_i) := 2^-K(i), even up to a constant, as Leon was initially hoping. With an additional step, we can make this true. Define a new UTM U’ that takes a code q and runs it to obtain an index i, then treats i as an index into the Friedberg numbering and simulates that machine (in other words, U’ decompresses a code for U). Now P_ap(\nu) for U’ matches 2^-K(\nu) up to a constant by the coding theorem.
Again: this entire argument depends on the unverified guess that there is a Friedberg numbering of l.s.c. semimeasures, which Marcus Hutter once suggested to me (but I think he does not believe it).
An idea I discussed with @Leon Lang that did not make it into the post is to look for a Friedberg numbering of the l.s.c. semimeasures, which could be used to control the weights P_ap. A Friedberg numbering for the recursively enumerable sets / partial recursive functions is (just) a unique effective numbering, which surprisingly enough does exist (though it is not admissable, meaning it can’t be translated to the standard numbering based on reasonable machine descriptions). If such a numbering existed, and U were chosen to implement it, then each l.s.c. semimeasure \nu would get prior weight (P_ap) exactly 2^(-l(p*)) where p* is the shortest program such that U(p*, _) generates the distribution of \nu.
l(p*) is one intuitive (but highly uncomputable) definition for K(\nu), but the normal definition is K(i) where i is the index of \nu (:= \nu_i) is some computable enumeration of l.s.c. semimeasures. Therefore, P_ap would still not match 2^-K(\nu_i) := 2^-K(i), even up to a constant, as Leon was initially hoping. With an additional step, we can make this true. Define a new UTM U’ that takes a code q and runs it to obtain an index i, then treats i as an index into the Friedberg numbering and simulates that machine (in other words, U’ decompresses a code for U). Now P_ap(\nu) for U’ matches 2^-K(\nu) up to a constant by the coding theorem.
Again: this entire argument depends on the unverified guess that there is a Friedberg numbering of l.s.c. semimeasures, which Marcus Hutter once suggested to me (but I think he does not believe it).