It should only require a hyperreal infinitesimal weight to maintain conservation of probability.
Doing arithmetic on infinities is not the same as doing infinite sequences of arithmetic. You can talk about a hyperreal-valued uniform prior, but can you actually do anything with it that you couldn’t do with an ordinary limit?
P(o%7CH_i)n%5E{-1})
The reasons that limit doesn’t suffice to specify a uniform prior are: (1) The result of the limit depends on the order of the list of hypotheses, which doesn’t sound very uniform to me (I don’t know if it’s worse than the choice of universal Turing machine in Solomonoff, but at least the pre-theoretic notion of simplicity comes with intuitions about which UTMs are simple). (2) For even more perverse orders, the limit doesn’t have to converge at all. (Even if utility is bounded, partial sums of EU can bounce around the bounded range forever.)
Hyperreal-valued expected utility doesn’t change (1). It does eliminate (2), but I think you have to sacrifice computability to do even that much: Construction of the hyperreals involves the axiom of choice, which prevents you from actually determining which real number is infinitesimally close to the hyperreal encoded by a given divergent sequence.
You wrote the above in regards to hyperreal infinities, “the hyperreal encoded by a given divergent sequence”. I’m under the impression that hyperreal infinitesimals are encoded by convergent sequences: specifically, sequences that converge to zero. The hyperreal [1, 1⁄2, 1⁄3, 1⁄4, …] is the one that corresponds to the limit you gave. Does that adequately dispel the computability issue you raised?
In any case, non-computability isn’t a major defect of the utilitarian prior vis-a-vis the also non-computable Solomonoff prior. It is an important caution, however.
Your first objection seems much more damaging to the idea of a utilitarian prior. Indeed, there seems little reason to expect max(U(o|Hi)) to vary in a systematic way with a useful enumeration of the hypotheses.
A non-constant sequence that converges to zero encodes an infinitesimal, and I think any infinitesimal has an encoding of that form. But a sequence that’s bounded in absolute value but doesn’t converge, e.g.
), also encodes some real plus some infinitesimal. It’s this latter kind that involves the axiom of choice, to put it in an equivalence class with some convergent sequence.
[1, 1⁄2, 1⁄3, 1⁄4, …] is the infinitesimal in the proposed definition of a uniform prior, but the hyperreal outcome of the expected utility calculation is
,%20{1\over%202}\sum_{i=1}%5E2U(o%7CH_i),%20{1\over%203}\sum_{i=1}%5E3U(o%7CH_i),%20...]) which might very well be the divergent kind.
Agreed that my first objection was more important.
Doing arithmetic on infinities is not the same as doing infinite sequences of arithmetic. You can talk about a hyperreal-valued uniform prior, but can you actually do anything with it that you couldn’t do with an ordinary limit?
The reasons that limit doesn’t suffice to specify a uniform prior are: (1) The result of the limit depends on the order of the list of hypotheses, which doesn’t sound very uniform to me (I don’t know if it’s worse than the choice of universal Turing machine in Solomonoff, but at least the pre-theoretic notion of simplicity comes with intuitions about which UTMs are simple). (2) For even more perverse orders, the limit doesn’t have to converge at all. (Even if utility is bounded, partial sums of EU can bounce around the bounded range forever.)
Hyperreal-valued expected utility doesn’t change (1). It does eliminate (2), but I think you have to sacrifice computability to do even that much: Construction of the hyperreals involves the axiom of choice, which prevents you from actually determining which real number is infinitesimally close to the hyperreal encoded by a given divergent sequence.
Thanks!
You wrote the above in regards to hyperreal infinities, “the hyperreal encoded by a given divergent sequence”. I’m under the impression that hyperreal infinitesimals are encoded by convergent sequences: specifically, sequences that converge to zero. The hyperreal [1, 1⁄2, 1⁄3, 1⁄4, …] is the one that corresponds to the limit you gave. Does that adequately dispel the computability issue you raised?
In any case, non-computability isn’t a major defect of the utilitarian prior vis-a-vis the also non-computable Solomonoff prior. It is an important caution, however.
Your first objection seems much more damaging to the idea of a utilitarian prior. Indeed, there seems little reason to expect max(U(o|Hi)) to vary in a systematic way with a useful enumeration of the hypotheses.
A non-constant sequence that converges to zero encodes an infinitesimal, and I think any infinitesimal has an encoding of that form. But a sequence that’s bounded in absolute value but doesn’t converge, e.g.
[1, 1⁄2, 1⁄3, 1⁄4, …] is the infinitesimal in the proposed definition of a uniform prior, but the hyperreal outcome of the expected utility calculation is
Agreed that my first objection was more important.