Person in a room: - 35 g of O2/hr from roomPerson in a room with a CO2 stripper: -35 g of O2/hr from roomHow does the presence of a CO2 stripper do anything at all to the oxygen amount in the air?
Do you think this problem is essentially different from "suppose Omega asks you for 10 bucks. You say no. Then Omega says "actually I flipped a fair coin that came up tails, if it had come up heads, I would have given you 100 dollars if I predicted you'd give me 10 dollars on tails"?(I think I can motivate "reconsider choosing heads" if you're like "yeah, this is just counterfactual mugging with belated notification of what situation you're in, and I'd pay up in that circumstance")
Maximin over outcomes would lead to the agent devoting all its efforts towards avoiding the worst outcomes, sacrificing overall utility, while maximin over expected value pushes towards policies that do acceptably on average in all of the environments that it may find itself in.Regarding "why listen to past me", I guess to answer this question I'd need to ask about your intuitions on Counterfactual mugging. What would you do if it's one-shot? What would you do if it's repeated? If you were told about the problem beforehand, would you pay money for a commitment mechanism to make future-you pay up the money if asked? (for +EV)
Yeah, looking back, I should probably fix the m- part and have the signs being consistent with the usual usage where it's a measure minus another one, instead of the addition of two signed measures, one a measure and one a negative measure. May be a bit of a pain to fix, though, the proof pages are extremely laggy to edit.Wikipedia's definition can be matched up with our definition by fixing a partial order where (m′,b′)≥(m,b) iff there's a (m∗,b∗) that's a sa-measure s.t. (m,b)+(m∗,b∗)=(m′,b′), and this generalizes to any closed convex cone. I lifted the definition of "positive functional" from Vanessa, though, you'd have to chat with her about it.We're talking about linear functions, not affine ones. (m,b)↦c(m(f)+b) is linear, not affine (regardless of f and c, as long as they're in C(X) and R, respectively). Observe that it maps the zero of M±(X)⊕R to 0.
We go to the trouble of sa-measures because it's possible to add a sa-measure to an a-measure, and get another a-measure where the expectation values of all the functions went up, while the new a-measure we landed at would be impossible to make by adding an a-measure to an a-measure.Basically, we've gotta use sa-measures for a clean formulation of "we added all the points we possibly could to this set", getting the canonical set in your equivalence class.Admittedly, you could intersect with the cone of a-measures again at the end (as we do in the next post) but then you wouldn't get the nice LF-duality tie-in.Adding the cone of a-measures instead would correspond to being able to take expectation values of continuous functions in [0,∞), instead of in [0,1], so I guess you could reformulate things this way, but IIRC the 0-1 normalization doesn't work as well (ie, there's no motive for why you're picking 1 as the thing to renormalize to 1 instead of, say, renormalizing 10 to 10). We've got a candidate other normalization for that case, but I remember being convinced that it doesn't work for belief functions, but for the Nirvana-as-1-reward-forever case, I remember getting really confused about the relative advantages of the two normalizations. And apparently, when working on the internal logic of infradistributions, this version of things works better.So, basically, if you drop sa-measures from consideration you don't get the nice LF-duality tie in and you don't have a nice way to express how upper completion works. And maybe you could work with a-measures and use upper completion w.r.t. a different cone and get a slightly different LF-duality, but then that would make normalization have to work differently and we haven't really cleaned up the picture of normalization in that case yet and how it interacts with everything else. I remember me and Vanessa switched our opinions like 3 times regarding which upper completion to use as we kept running across stuff and going "wait no, I take back my old opinion, the other one works better with this".
Can you elaborate on what you meant by locally distinguishing between hypotheses?
If hospitals are overwhelmed, it's valuable to have a component of the hospital treatment plan for pneumonia on-hand to treat either yourself or others who have it especially bad. One of these is oxygen concentrators, which are not sold out yet and are ~$400 on Amazon. This doesn't deal with especially severe cases, but for cases which fall in the "shortness of breath, low blood oxygen" class without further medical complications, it'd probably be useful if you can't or don't want to go to a hospital due to overload. https://www.who.int/publications-detail/clinical-management-of-severe-acute-respiratory-infection-when-novel-coronavirus-(ncov)-infection-is-suspected mentions oxygen treatment as the first thing to do for low blood oxygen levels.
I found a paper about this exact sort of thing. Escardo and Olivia call that type signature a "selection functional", and the type signature (A→B)→B is called a "quantification functional", and there's several interesting things you can do with them, like combining multiple selection functionals into one in a way that looks reminiscent of game theory. (ie, if ϵ has type signature (A→C)→A, and δ has type signature (B→C)→B, then ϵ⊗δ has type signature ((A×B)→C)→(A×B).
Oh, I see what the issue is. Propositional tautology given A means A⊢pcϕ, not A⊢ϕ. So yeah, when A is a boolean that is equivalent to ⊥ via boolean logic alone, we can't use that A for the exact reason you said, but if A isn't equivalent to ⊥ via boolean logic alone (although it may be possible to infer ⊥ by other means), then the denominator isn't necessarily small.
Yup, a monoid, because ϕ∨⊥=ϕ and A∪∅=A, so it acts as an identitity element, and we don't care about the order. Nice catch.
You're also correct about what propositional tautology given A means.