A physicalist hypothesis is a pair (Φ,Θ), where Φ is a finite[4:2] set representing the physical states of the universe and Θ∈□(Γ×Φ) represents a joint belief about computations and physics. [...] Our agent will have a prior over such hypotheses, ranging over different Φ.
I am confused what the state space Φ is adding to your formalism and how it is supposed to solve the ontology identification problem. Based on what I understood, if I want to use this for inference, I have this prior ξ∈□c(Φ,Θ), and now I can use the bridge transform to project phi out again to evaluate my loss in different counterfactuals. But when looking at your loss function, it seems like most of the hard work is actually done in your relation C∈elΓ that determines which universes are consistent, but its definition does not seem to depend on Φ. How is that different from having a prior that is just over ξ∈□c(Γ) and taking the loss, if Φ is projected out anyway and thus not involved?
First, the notation ξ∈□c(Φ,Θ) makes no sense. The prior is over hypotheses, each of which is an element of □(Γ×Φ). Θ is the notation used to denote a single hypothesis.
Second, having a prior just over Γ doesn’t work since both the loss function and the counterfactuals depend on 2Γ×Γ.
Third, the reason we don’t just start with a prior over 2Γ×Γ, is because it’s important which prior we have. Arguably, the correct prior is the image of a simplicity prior over physicalist hypotheses by the bridge transform. But, come to think about it, it might be about the same as having a simplicity prior over 2Γ×Γ, where each hypothesis is constrained to be invariant under the bridge transform (thanks to Proposition 2.8). So, maybe we can reformulate the framework to get rid of Φ (but not of the bridge transform). Then again, finding the “ultimate prior” for general intelligence is a big open problem, and maybe in the end we will need to specify it with the help of Φ.
Fourth, I wouldn’t say that Φ is supposed to solve the ontology identification problem. The way IBP solves the ontology identification problem is by asserting that 2Γ×Γ is the correct ontology. And then there are tricks how to translate between other ontologies and this ontology (which is what section 3 is about).
I am confused what the state space Φ is adding to your formalism and how it is supposed to solve the ontology identification problem. Based on what I understood, if I want to use this for inference, I have this prior ξ∈□c(Φ,Θ), and now I can use the bridge transform to project phi out again to evaluate my loss in different counterfactuals. But when looking at your loss function, it seems like most of the hard work is actually done in your relation C∈elΓ that determines which universes are consistent, but its definition does not seem to depend on Φ. How is that different from having a prior that is just over ξ∈□c(Γ) and taking the loss, if Φ is projected out anyway and thus not involved?
First, the notation ξ∈□c(Φ,Θ) makes no sense. The prior is over hypotheses, each of which is an element of □(Γ×Φ). Θ is the notation used to denote a single hypothesis.
Second, having a prior just over Γ doesn’t work since both the loss function and the counterfactuals depend on 2Γ×Γ.
Third, the reason we don’t just start with a prior over 2Γ×Γ, is because it’s important which prior we have. Arguably, the correct prior is the image of a simplicity prior over physicalist hypotheses by the bridge transform. But, come to think about it, it might be about the same as having a simplicity prior over 2Γ×Γ, where each hypothesis is constrained to be invariant under the bridge transform (thanks to Proposition 2.8). So, maybe we can reformulate the framework to get rid of Φ (but not of the bridge transform). Then again, finding the “ultimate prior” for general intelligence is a big open problem, and maybe in the end we will need to specify it with the help of Φ.
Fourth, I wouldn’t say that Φ is supposed to solve the ontology identification problem. The way IBP solves the ontology identification problem is by asserting that 2Γ×Γ is the correct ontology. And then there are tricks how to translate between other ontologies and this ontology (which is what section 3 is about).