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As an outgrowth of this discussion I tried submitting a question to Ask HN but it went over the character limit, so for want of a better place to put it, maybe

loup-vaillant 14y ago
I'm not sure what to make of this… More precisely, I'm not sure what your fractions on the left actually mean. Personally, I'm tempted to throw "undefined" at each row that does not sum to 1, leaving only "1/0", ".5/.5", and "0/1". I haven't heard about any distinction between "subjectivists" and "objectivists" Bayesians. I'd say my own position is a little bit of both: First, background knowledge fully constraint the world view. That means that two rational agents whose knowledges are the same must hold the same beliefs. More practically, two persons having the same relevant knowledge about any given question should believe the same answer (the difficulty here is to sort out what's relevant). Second, agents generally don't have the same background knowledge. Nothing stops us for knowing different relevant information about any given subject. So of course we won't hold the same beliefs. Fuzzy. Ah, fuzzy… the more I look at it, the less sense it makes. Strictly speaking, there is no such thing as "half truth". A proposition is binary: either true or false. Beliefs on the other hand are not binary: a belief is a proposition plus a probability. If I say the bottle is more than 75% full with probability 84%, then I'm 16% right if it really is only half full. Quite wrong, but not hopelessly so. With probability functions and utility function, we can make the same set of decisions we could have made with fuzzy logic, which looks like it mixes the two. I personally prefer the modular approach.
- tunesmith 14y ago
  The fractions are supposed to represent partial degrees of truth. 0.5 is just an example, could be 0.4, 0.6 - anything other than absolute true or false. There are examples of probability systems that try to factor in uncertainty, beyond truth belief or false belief. I think it was possibility theory that was talking about a 0.5 value assigned to "the cat is alive", a 0.2 value assigned to "the cat is dead", and a 0.3 uncertainty - then it would be 0.8 plausibility/possibility of alive, but only 0.5 probability. The main distinction I saw about subjectivist vs objectivist is that the objectivist still communicates in terms of degree of belief, but the belief is that it is based on objective empirical data and so therefore it's not an expression of personal belief - more a measure of belief for the entire system or knowledge base. More relevant for things like machine learning rather than updating ones own personal beliefs. (Maybe this just assumes common priors.) I've gone round and round on fuzzy because some people talk about fuzzy as if it is degrees-of-belief, like the bayesian interpretation. But I think it's different - I mean, if I'm partly correct on describing a concept, you wouldn't say that I am right 50 times out of 100, or that you're 50% certain I'm right, right? You'd say I'm half-right. I'll have to think about probability combined with utility function. I'm not sure how to make it square with fuzzy math. It seems you'd be 16% right whether the bottle was half-full, quarter-full, or empty.

loup-vaillant 14y ago

I'm not sure what to make of this… More precisely, I'm not sure what your fractions on the left actually mean. Personally, I'm tempted to throw "undefined" at each row that does not sum to 1, leaving only "1/0", ".5/.5", and "0/1". I haven't heard about any distinction between "subjectivists" and "objectivists" Bayesians. I'd say my own position is a little bit of both: First, background knowledge fully constraint the world view. That means that two rational agents whose knowledges are the same must hold the same beliefs. More practically, two persons having the same relevant knowledge about any given question should believe the same answer (the difficulty here is to sort out what's relevant). Second, agents generally don't have the same background knowledge. Nothing stops us for knowing different relevant information about any given subject. So of course we won't hold the same beliefs. Fuzzy. Ah, fuzzy… the more I look at it, the less sense it makes. Strictly speaking, there is no such thing as "half truth". A proposition is binary: either true or false. Beliefs on the other hand are not binary: a belief is a proposition plus a probability. If I say the bottle is more than 75% full with probability 84%, then I'm 16% right if it really is only half full. Quite wrong, but not hopelessly so. With probability functions and utility function, we can make the same set of decisions we could have made with fuzzy logic, which looks like it mixes the two. I personally prefer the modular approach.

tunesmith 14y ago
The fractions are supposed to represent partial degrees of truth. 0.5 is just an example, could be 0.4, 0.6 - anything other than absolute true or false. There are examples of probability systems that try to factor in uncertainty, beyond truth belief or false belief. I think it was possibility theory that was talking about a 0.5 value assigned to "the cat is alive", a 0.2 value assigned to "the cat is dead", and a 0.3 uncertainty - then it would be 0.8 plausibility/possibility of alive, but only 0.5 probability. The main distinction I saw about subjectivist vs objectivist is that the objectivist still communicates in terms of degree of belief, but the belief is that it is based on objective empirical data and so therefore it's not an expression of personal belief - more a measure of belief for the entire system or knowledge base. More relevant for things like machine learning rather than updating ones own personal beliefs. (Maybe this just assumes common priors.) I've gone round and round on fuzzy because some people talk about fuzzy as if it is degrees-of-belief, like the bayesian interpretation. But I think it's different - I mean, if I'm partly correct on describing a concept, you wouldn't say that I am right 50 times out of 100, or that you're 50% certain I'm right, right? You'd say I'm half-right. I'll have to think about probability combined with utility function. I'm not sure how to make it square with fuzzy math. It seems you'd be 16% right whether the bottle was half-full, quarter-full, or empty.