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Sunday, December 8th 2019
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40BrienneYudkowsky1moSuppose you wanted to improve your social relationships on the community level. (I think of this as “my ability to take refuge in the sangha”.) What questions might you answer now, and then again in one year, to track your progress? Here’s what’s come to mind for me so far. I’m probably missing a lot and would really like your help mapping things out. I think it’s a part of the territory I can only just barely perceive at my current level of development. * If something tragic happened to you, such as a car crash that partially paralyzed you or the death of a loved one, how many people can you name whom you'd find it easy and natural to ask for help with figuring out your life afterward? * For how many people is it the case that if they were hospitalized for at least a week you would visit them in the hospital? * Over the past month, how lonely have you felt? * In the past two weeks, how often have you collaborated with someone outside of work? * To what degree do you feel like your friends have your back? * Describe the roll of community in your life. * How do you feel as you try to describe the roll of community in your life? * When's the last time you got angry with someone and confronted them one on one as a result? * When's the last time you apologized to someone? * How strong is your sense that you're building something of personal value with the people around you? * When's the last time you spent more than ten minutes on something that felt motivated by gratitude? * When a big change happens in your life, such as loosing your job or having a baby, how motivated do you feel to share the experience with others? * When you feel motivated to share an experience with others, how satisfied do you tend to be with your attempts to do that? * Do you know the love languages of your five closest friends? To what extent does that influence how you behave toward them? * Does it seem to you that your friends know your love
7AlexMennen1moTheorem: Fuzzy beliefs (as in https://www.alignmentforum.org/posts/Ajcq9xWi2fmgn8RBJ/the-credit-assignment-problem#X6fFvAHkxCPmQYB6v [https://www.alignmentforum.org/posts/Ajcq9xWi2fmgn8RBJ/the-credit-assignment-problem#X6fFvAHkxCPmQYB6v] ) form a continuous DCPO. (At least I'm pretty sure this is true. I've only given proof sketches so far) The relevant definitions: A fuzzy belief over a set X is a concave function ϕ:ΔX→[0,1] such that sup(ϕ)=1 (where ΔX is the space of probability distributions on X). Fuzzy beliefs are partially ordered by ϕ≤ψ⟺∀μ∈ΔX:ϕ(μ)≥ψ(μ) . The inequalities reverse because we want to think of "more specific"/"less fuzzy" beliefs as "greater", and these are the functions with lower values; the most specific/least fuzzy beliefs are ordinary probability distributions, which are represented as the concave hull of the function assigning 1 to that probability distribution and 0 to all others; these should be the maximal fuzzy beliefs. Note that, because of the order-reversal, the supremum of a set of functions refers to their pointwise infimum. A DCPO (directed-complete partial order) is a partial order in which every directed subset has a supremum. In a DCPO, define x<<y to mean that for every directed set D with supD≥y, ∃d∈D such that d≥x. A DCPO is continuous if for every y , y=sup{x∣x<<y}. Lemma: Fuzzy beliefs are a DCPO. Proof sketch: Given a directed set D , (supD)(μ)=min{d(μ)∣d∈D} is convex, and {μ∣(supD)(μ)=1}=⋂d∈D{μ∣d(μ)=1}. Each of the sets in that intersection are non-empty, hence so are finite intersections of them since D is directed, and hence so is the whole intersection since ΔX is compact. Lemma: ϕ<<ψ iff {μ∣ψ(μ)=1} is contained in the interior of {μ∣ϕ(μ)=1} and for every μ such that ψ(μ)≠1, ϕ(μ)>ψ(μ). Proof sketch: If supD≥ψ, then ⋂d∈D{μ∣d(μ)=1}⊆{μ∣ψ(μ)=1} , so by compactness of ΔX and directedness of D, there should be d∈D such that {μ∣d(μ)=1}⊆int({μ∣ϕ(μ)=1}). Similarly, for each μ such that ψ(μ)≠1, there should be dμ∈D s