For the sake of potential readers, a (full) distribution over X is some γ:X→[0,1] with finite support and ∑x∈Xγ(x)=1, whereas a subdistribution over X is some γ:X→[0,1] with finite support and ∑x∈Xγ(x)≤1. Note that a subdistribution γ over X is equivalent to a full distribution over X+1, where X+1 is the disjoint union of X with some additional element, so the subdistribution monad can be written Δ(−+1).
I am not at all convinced by the interpretation of (−+2) here as terminating a game with a reward for the adversary or the agent. My interpretation of the distinguished element ⊥ in (−+1) is not that it represents a special state in which the game is over, but rather a special state in which there is a contradiction between some of one’s assumptions/observations.
Doesn’t the Nirvana Trick basically say that these two interpretations are equivalent?
Let (−+2) be X↦X+{0,1} and let (−+1) be X↦X+{0}. We can interpret ∨ as possibility, 0 as a hypothesis consistent with no observations, and 1 as a hypothesis consistent with all observations.
Alternatively, we can interpret ∨ as the free choice made by an adversary, 0 as “the game terminates and our agent receives minimal disutility”, and 1 as “the game terminates and our agent receives maximal disutility”. These two interpretations are algebraically equivalent, i.e.(∨,0,1) is a topped and bottomed semilattice.
Unless I’m mistaken, both P+f∘Δ∘(−+2) and P+f∘Δ∘(−+1) demand that the agent may have the hypothesis “I am certain that I will receive minimal disutility”, which is necessary for the Nirvana Trick. But P+f∘Δ∘(−+2) also demands that the agent may have the hypothesis “I am certain that I will receive maximal disutility”. The first gives bounded infrabayesian monad and the second gives unbounded infrabayesian monad. Note that Diffractor uses P+f∘Δ∘(−+2) in Infra-Miscellanea Section 2.
I agree that each of (−+1) and (−+2) has two algebraically equivalent interpretations, as you say, where one is about inconsistency and the other is about inferiority for the adversary. (I hadn’t noticed that).
The (−+2) variant still seems somewhat irregular to me; even though Diffractor does use it in Infra-Miscellanea Section 2, I wouldn’t select it as “the” infrabayesian monad. I’m also confused about which one you’re calling unbounded. It seems to me like the (−+2) variant is bounded (on both sides) whereas the (−+1) variant is bounded on one side, and neither is really unbounded. (Being bounded on at least one side is of course necessary for being consistent with infinite ethics.)
For the sake of potential readers, a (full) distribution over X is some γ:X→[0,1] with finite support and ∑x∈Xγ(x)=1, whereas a subdistribution over X is some γ:X→[0,1] with finite support and ∑x∈Xγ(x)≤1. Note that a subdistribution γ over X is equivalent to a full distribution over X+1, where X+1 is the disjoint union of X with some additional element, so the subdistribution monad can be written Δ(−+1).
Doesn’t the Nirvana Trick basically say that these two interpretations are equivalent?
Let (−+2) be X↦X+{0,1} and let (−+1) be X↦X+{0}. We can interpret ∨ as possibility, 0 as a hypothesis consistent with no observations, and 1 as a hypothesis consistent with all observations.
Alternatively, we can interpret ∨ as the free choice made by an adversary, 0 as “the game terminates and our agent receives minimal disutility”, and 1 as “the game terminates and our agent receives maximal disutility”. These two interpretations are algebraically equivalent, i.e.(∨,0,1) is a topped and bottomed semilattice.
Unless I’m mistaken, both P+f∘Δ∘(−+2) and P+f∘Δ∘(−+1) demand that the agent may have the hypothesis “I am certain that I will receive minimal disutility”, which is necessary for the Nirvana Trick. But P+f∘Δ∘(−+2) also demands that the agent may have the hypothesis “I am certain that I will receive maximal disutility”. The first gives bounded infrabayesian monad and the second gives unbounded infrabayesian monad. Note that Diffractor uses P+f∘Δ∘(−+2) in Infra-Miscellanea Section 2.
I agree that each of (−+1) and (−+2) has two algebraically equivalent interpretations, as you say, where one is about inconsistency and the other is about inferiority for the adversary. (I hadn’t noticed that).
The (−+2) variant still seems somewhat irregular to me; even though Diffractor does use it in Infra-Miscellanea Section 2, I wouldn’t select it as “the” infrabayesian monad. I’m also confused about which one you’re calling unbounded. It seems to me like the (−+2) variant is bounded (on both sides) whereas the (−+1) variant is bounded on one side, and neither is really unbounded. (Being bounded on at least one side is of course necessary for being consistent with infinite ethics.)