Kosoy’s infrabayesian monad □ is given by P+∘Δ∘(−+2)
There are a few different varieties of infrabayesian belief-state, but I currently favour the one which is called “homogeneous ultracontributions”, which is “non-empty topologically-closed ⊥–closed convex sets of subdistributions”, thus almost exactly the same as Mio-Sarkis-Vignudelli’s “non-empty finitely-generated ⊥–closed convex sets of subdistributions monad” (Definition 36 of this paper), with the difference being essentially that it’s presentable, but it’s much more like P+f∘Δ∘(−+1) than P+f∘Δ∘(−+2).
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. This is very useful for modelling Bayesian updates (Evidential Decision Theory via Partial Markov Categories, sections 3.5-3.6), in which some variable X is observed to satisfy a certain predicate q: this can be modelled by applying the predicate in the form q:X→□{∗} where q(x)=⊥ means the predicate is false, and q(x)=∗ means it is true. But I don’t think there is a dual to logical inconsistency, other than the full set of all possible subdistributions on the state space. It is certainly not the same type of “failure” as losing a game.
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.)
There are a few different varieties of infrabayesian belief-state, but I currently favour the one which is called “homogeneous ultracontributions”, which is “non-empty topologically-closed ⊥–closed convex sets of subdistributions”, thus almost exactly the same as Mio-Sarkis-Vignudelli’s “non-empty finitely-generated ⊥–closed convex sets of subdistributions monad” (Definition 36 of this paper), with the difference being essentially that it’s presentable, but it’s much more like P+f∘Δ∘(−+1) than P+f∘Δ∘(−+2).
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. This is very useful for modelling Bayesian updates (Evidential Decision Theory via Partial Markov Categories, sections 3.5-3.6), in which some variable X is observed to satisfy a certain predicate q: this can be modelled by applying the predicate in the form q:X→□{∗} where q(x)=⊥ means the predicate is false, and q(x)=∗ means it is true. But I don’t think there is a dual to logical inconsistency, other than the full set of all possible subdistributions on the state space. It is certainly not the same type of “failure” as losing a game.
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.)