I’m using some of the terminology I suggested here.
A factoring is a set of questions such that each signature of possible answers identifies a unique element. In 20 questions, you can tailor the questions depending on the answers to previous questions, and ultimately each element will have a bitstring signature depending on the history of yesses and nos. I guess you can define the question to include xors with previous questions, so that it effectively changes depending on the answers to others. But it’s sometimes useful that the bitstrings are allowed to have different length. It feels like an unfortunate fact that when constructing a factoring for 7 elements, you’re forced to use the factoring {”Okay, well, which element is it?”}, just because you don’t want to have to answer a different number of questions for different elements. Is this a real cost? Or do we only ever construct cases where it’s not?
In the directed graph of subsets, with edges corresponding to the subset relation, why not consider arbitrary subtrees? For example, for the set of 7 elements, we might have {{0, 1, 2}, {{3, 4}, {5, 6}}}. (I’m not writing it out as a tree, but that contains all the information). This corresponds to the sequence of questions: “is it less than 3?”, [if yes] “is it 0, 1, or 2?”, [if no], “is it less than 5?”, “is it even?” Allowing different numbers of questions and different numbers of answers seems to give some extra power here. Is it meaningful?
I think that the answers to both the concern about 7 elements, and the desire to have questions depend of previous questions come out of thinking about FFS models, rather than FFS.
If you want to have 7 elements in Ω, that just means you will probably have more than 7 elements in S.
If I want to model a situation where some questions I ask depend on other questions, I can just make a big FFS that asks all the questions, and have the model hide some of the answers.
For example, Let’s say I flip a biased coin, and then if heads I roll a biased 6 sided die, and if tails I roll a biased 10 sided die. There are 16 outcomes in Ω.
I can build a 3 dimensional factored set 2x6x10, which I will imagine as sitting on my table with height 2. heads is on the bottom, and tails is on the top.f:S→Ω will then merge together the rows on the bottom, and the columns on the top, so it will look a little like the game Jenga.
In this way, I am imagining there is some hidden data about each world in which I get heads and roll the 6 sided die, which is the answer to the question “what would have happened if I rolled the 10 sided die. Adding in all this counterfactual data gives a latent structure of 120 possible worlds, even though we can only distinguish 16 possible worlds.
I’m using some of the terminology I suggested here.
A factoring is a set of questions such that each signature of possible answers identifies a unique element. In 20 questions, you can tailor the questions depending on the answers to previous questions, and ultimately each element will have a bitstring signature depending on the history of yesses and nos. I guess you can define the question to include xors with previous questions, so that it effectively changes depending on the answers to others. But it’s sometimes useful that the bitstrings are allowed to have different length. It feels like an unfortunate fact that when constructing a factoring for 7 elements, you’re forced to use the factoring {”Okay, well, which element is it?”}, just because you don’t want to have to answer a different number of questions for different elements. Is this a real cost? Or do we only ever construct cases where it’s not?
In the directed graph of subsets, with edges corresponding to the subset relation, why not consider arbitrary subtrees? For example, for the set of 7 elements, we might have {{0, 1, 2}, {{3, 4}, {5, 6}}}. (I’m not writing it out as a tree, but that contains all the information). This corresponds to the sequence of questions: “is it less than 3?”, [if yes] “is it 0, 1, or 2?”, [if no], “is it less than 5?”, “is it even?” Allowing different numbers of questions and different numbers of answers seems to give some extra power here. Is it meaningful?
I think that the answers to both the concern about 7 elements, and the desire to have questions depend of previous questions come out of thinking about FFS models, rather than FFS.
If you want to have 7 elements in Ω, that just means you will probably have more than 7 elements in S.
If I want to model a situation where some questions I ask depend on other questions, I can just make a big FFS that asks all the questions, and have the model hide some of the answers.
For example, Let’s say I flip a biased coin, and then if heads I roll a biased 6 sided die, and if tails I roll a biased 10 sided die. There are 16 outcomes in Ω.
I can build a 3 dimensional factored set 2x6x10, which I will imagine as sitting on my table with height 2. heads is on the bottom, and tails is on the top.f:S→Ω will then merge together the rows on the bottom, and the columns on the top, so it will look a little like the game Jenga.
In this way, I am imagining there is some hidden data about each world in which I get heads and roll the 6 sided die, which is the answer to the question “what would have happened if I rolled the 10 sided die. Adding in all this counterfactual data gives a latent structure of 120 possible worlds, even though we can only distinguish 16 possible worlds.