Thanks for the link! What I don’t understand is how this works in the context of empirical and logical uncertainty. Also, it’s unclear to me how this approach relates to Bayesian conditioning. E.g. if the sentence “if a holds, than o holds” is true, doesn’t this also mean that P(o|a)=1? In that sense, proof-based UDT would just be an elaborate specification of how to assign these conditional probabilities “from the viewpoint of the original position”, so with updatelessness, and in the context of full logical inference and knowledge of the world, including knowledge about one’s own decision algorithm. I see how this is useful, but don’t understand how it would at any point contradict normal Bayesian conditioning.
As to your first question: if we ignore problems that involve updatelessness (or if we just stipulate that EDT always had the opportunity to precommit), I haven’t been able to find any formally specified problems where EDT and UDT diverge.
I think Caspar Oesterheld’s and my flavor of EDT would be ordinary EDT with some version of updatelessness. I’m not sure if this works, but if it turns out to be identical to UDT, then I’m not sure which of the two is the better specified or easier to formalize one. According to the language in Arbital’s LDT article, my EDT would differ from UDT only insofar as instead of some logical conditioning, we use ordinary Bayesian conditioning. So (staying in the Arbital framework), it could look something like this (P stands for whatever prior probability distribution you care about):
Also, it’s unclear to me how this approach relates to Bayesian conditioning.
To me, proof-based UDT is a simple framework that includes probabilistic/Bayesian reasoning as a special case. For example, if the world is deterministic except for a single coinflip, you specify a preference ordering on pairs of outcomes of two deterministic worlds. Fairness or non-fairness of the coinflip will be encoded into the ordering, so the decision can be based on completely deterministic reasoning. All probabilistic situations can be recast in this way. That’s what UDT folks mean by “probability as caring”.
It’s really cool that UDT lets you encode any setup with probability, prediction, precommitment etc. into a few (complicated and self-referential) sentences in PA or GL that are guaranteed to have truth values. And since GL is decidable, you can even write a program that will solve all such problems for you.
Thanks for the link! What I don’t understand is how this works in the context of empirical and logical uncertainty. Also, it’s unclear to me how this approach relates to Bayesian conditioning. E.g. if the sentence “if a holds, than o holds” is true, doesn’t this also mean that P(o|a)=1? In that sense, proof-based UDT would just be an elaborate specification of how to assign these conditional probabilities “from the viewpoint of the original position”, so with updatelessness, and in the context of full logical inference and knowledge of the world, including knowledge about one’s own decision algorithm. I see how this is useful, but don’t understand how it would at any point contradict normal Bayesian conditioning.
As to your first question: if we ignore problems that involve updatelessness (or if we just stipulate that EDT always had the opportunity to precommit), I haven’t been able to find any formally specified problems where EDT and UDT diverge.
I think Caspar Oesterheld’s and my flavor of EDT would be ordinary EDT with some version of updatelessness. I’m not sure if this works, but if it turns out to be identical to UDT, then I’m not sure which of the two is the better specified or easier to formalize one. According to the language in Arbital’s LDT article, my EDT would differ from UDT only insofar as instead of some logical conditioning, we use ordinary Bayesian conditioning. So (staying in the Arbital framework), it could look something like this (P stands for whatever prior probability distribution you care about):
(argmaxπx∈Π ∑oi∈OU(oi)⋅P(oi|πx))(s)
To me, proof-based UDT is a simple framework that includes probabilistic/Bayesian reasoning as a special case. For example, if the world is deterministic except for a single coinflip, you specify a preference ordering on pairs of outcomes of two deterministic worlds. Fairness or non-fairness of the coinflip will be encoded into the ordering, so the decision can be based on completely deterministic reasoning. All probabilistic situations can be recast in this way. That’s what UDT folks mean by “probability as caring”.
It’s really cool that UDT lets you encode any setup with probability, prediction, precommitment etc. into a few (complicated and self-referential) sentences in PA or GL that are guaranteed to have truth values. And since GL is decidable, you can even write a program that will solve all such problems for you.