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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

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
  Okay, so I got your fractions mostly right. Well, when you use a computer to make a probabilistic calculation, you have at some point to feed it with the relevant information, or it won't know what to make of the data. The data alone is not enough. And if you are absolutely certain your program is (i) correct, and (ii) fed with all the relevant information you know about, then you should definitely believe the result. (Of course, this absolute certainty is not attainable, so any sufficiently surprising result should lead you to think that something went wrong.) Assuming common priors on the other hand, seems unreasonable, unless we're only talking about "fully uninformed" priors such as those based on Kolmogorov complexity. Yes, I would say you're half right. The example of the temperature given by the Wikipedia is really good. The "degree of coldness" is a worthy notion. What bothers me is that fuzzy logicians don't seem to run probability distributions over those degrees of truths. I mean, I can surely make up a probability distribution on the outside temperature for tomorrow at 10AM. Assuming that temperature maps to degrees of truth to the sentence "it's cold" (say, 100% true below 10°C, 0% true beyond 20°C, and a linear interpolation in between), then I naturally come up with a probability distribution over the degrees of truth (I should get a Dirac spike at both extremities). Your last one is exactly right. I'm only 16% from empty to 75% full (excluded). That's the punishment I get for not producing a theory of the fullness of the bottle which describes my actual beliefs about it. I assume that your ultimate goal in life is to make decisions, or to create a machine that will take decisions. To make sound decisions, you need to asses their consequences. Take the example of the bottle: I need a full bottle of water to hike today. Let's assume for the sake of the argument that it is either empty, or full. If it's empty, I need to re-fill it, or I would faint from the thirst (not good). If it's full, checking is only a nuisance (not good either). Now the question is, how can I minimize my discomfort, on average? Well I need two things: first, a probability distribution. Let's say I'm pretty sure I filled the bottle yesterday, let's call it a 90% chance. Second, a utility function. Baseline, the bottle is full, and I did not bother checking it, utility zero. Checking costs me 1 utility point, and fainting from thirst costs me 50 points (let's assume my brain is actually capable of performing those utility assessments). Should I check the bottle? Oh yes. Not checking means a 10% chance of losing 50 points, or 5 on average. Checking means a certainty of losing only one point, which costs much less. Conclusion: check your water supply before you go hiking. And re-check just to be sure. Now this problem didn't call for fuzzy logic. We can however come up with a full probability distribution over the quantity of water in the bottle instead of just two possibilities. From then, fuzzy logic should be able to naturally step in. But frankly, I prefer to run a a separate (dis)utility function over the state of thirst that the lack of water will provoke (from light thirst to actually faint), and combine it with my probability distribution to make my decision. (Though at that point, I'd rather just check than frying my brain trying to solve this cost-benefit calculation.)
Jach 14y ago
Fuzzy set theory makes sense again when you realize it's not about classifying uncertainty or partial truth at all, but merely denoting partial set membership. Goertzel's et aliorum PLN book and framework that I mentioned before use fuzzy values in exactly that way actually. There's a brief high-level overview on the differences between PLN and other approaches to uncertain inference early on in the book. In PLN, all of these four sentences are uniquely distinguished, whereas in a fuzzy framework it's not so clear what's going on beyond the first sentence: "Jim belongs to degree 0.6 to the fuzzy set of tall people.", "Jim shares 0.6 of the properties shared by people belonging to the set of tall people (where the different properties may be weighted).", "Jim has a 0.6 chance of being judged as belonging to the set of tall people, once more information about Jim is obtained (where this may be weighted as to the degree of membership that is expected to be estimated once the additional information is obtained).", and "Jim has an overall 0.6 amount of tallness, defined as a weighted average of extensional and intensional information."
- loup-vaillant 14y ago
  Yes, it does make sense.

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
Okay, so I got your fractions mostly right. Well, when you use a computer to make a probabilistic calculation, you have at some point to feed it with the relevant information, or it won't know what to make of the data. The data alone is not enough. And if you are absolutely certain your program is (i) correct, and (ii) fed with all the relevant information you know about, then you should definitely believe the result. (Of course, this absolute certainty is not attainable, so any sufficiently surprising result should lead you to think that something went wrong.) Assuming common priors on the other hand, seems unreasonable, unless we're only talking about "fully uninformed" priors such as those based on Kolmogorov complexity. Yes, I would say you're half right. The example of the temperature given by the Wikipedia is really good. The "degree of coldness" is a worthy notion. What bothers me is that fuzzy logicians don't seem to run probability distributions over those degrees of truths. I mean, I can surely make up a probability distribution on the outside temperature for tomorrow at 10AM. Assuming that temperature maps to degrees of truth to the sentence "it's cold" (say, 100% true below 10°C, 0% true beyond 20°C, and a linear interpolation in between), then I naturally come up with a probability distribution over the degrees of truth (I should get a Dirac spike at both extremities). Your last one is exactly right. I'm only 16% from empty to 75% full (excluded). That's the punishment I get for not producing a theory of the fullness of the bottle which describes my actual beliefs about it. I assume that your ultimate goal in life is to make decisions, or to create a machine that will take decisions. To make sound decisions, you need to asses their consequences. Take the example of the bottle: I need a full bottle of water to hike today. Let's assume for the sake of the argument that it is either empty, or full. If it's empty, I need to re-fill it, or I would faint from the thirst (not good). If it's full, checking is only a nuisance (not good either). Now the question is, how can I minimize my discomfort, on average? Well I need two things: first, a probability distribution. Let's say I'm pretty sure I filled the bottle yesterday, let's call it a 90% chance. Second, a utility function. Baseline, the bottle is full, and I did not bother checking it, utility zero. Checking costs me 1 utility point, and fainting from thirst costs me 50 points (let's assume my brain is actually capable of performing those utility assessments). Should I check the bottle? Oh yes. Not checking means a 10% chance of losing 50 points, or 5 on average. Checking means a certainty of losing only one point, which costs much less. Conclusion: check your water supply before you go hiking. And re-check just to be sure. Now this problem didn't call for fuzzy logic. We can however come up with a full probability distribution over the quantity of water in the bottle instead of just two possibilities. From then, fuzzy logic should be able to naturally step in. But frankly, I prefer to run a a separate (dis)utility function over the state of thirst that the lack of water will provoke (from light thirst to actually faint), and combine it with my probability distribution to make my decision. (Though at that point, I'd rather just check than frying my brain trying to solve this cost-benefit calculation.)

Jach 14y ago

Fuzzy set theory makes sense again when you realize it's not about classifying uncertainty or partial truth at all, but merely denoting partial set membership. Goertzel's et aliorum PLN book and framework that I mentioned before use fuzzy values in exactly that way actually. There's a brief high-level overview on the differences between PLN and other approaches to uncertain inference early on in the book. In PLN, all of these four sentences are uniquely distinguished, whereas in a fuzzy framework it's not so clear what's going on beyond the first sentence: "Jim belongs to degree 0.6 to the fuzzy set of tall people.", "Jim shares 0.6 of the properties shared by people belonging to the set of tall people (where the different properties may be weighted).", "Jim has a 0.6 chance of being judged as belonging to the set of tall people, once more information about Jim is obtained (where this may be weighted as to the degree of membership that is expected to be estimated once the additional information is obtained).", and "Jim has an overall 0.6 amount of tallness, defined as a weighted average of extensional and intensional information."

loup-vaillant 14y ago
Yes, it does make sense.