I tell you I believe X with 54% certainty. Who knows, that number could have been generated in a completely bogus way. But however I got here, this is where I am. There are bets about X that I will and won’t take, and guess what, that’s my cutoff probability right there. And by the way, now I have communicated to you where I am, in a way that does not further compound the error.
Meaningless is a very strong word.
In the face of such uncertainty, it could feel natural to take shelter in the idea of “inherent vagueness”...but this is reality, and we place our bets with real dollars and cents, and all the uncertainty in the world collapses to a number in the face of the expectation operator.
So why stop there? If you can justify 54%, then why not go further and calculate a dozen or two more significant digits, and stand behind them all with unshaken resolve?
You can, of course. For most situations, the effort is not worth the trade-off. But making a distinction between 1%, 25%, 50%. 75%. and 99% often is.
You can (at least formally) put error bars on the quantities that go into a Bayesian calculation. The problem, of course, is that error bars are short-hand for a distribution of possible values, and it’s not obvious what a distribution of probabilities means or should mean. Everything operational about probability functions is fully captured by their full set of expectation values, so this is no different than just immediately taking the mean, right?
Well, no. The uncertainties are a higher level model that not only makes predictions, but also calibrates how much these predictions are likely to move given new data.
It seems to me that this is somewhat related to the problem of logical uncertainty.
Again, meaningless is a very strong word, and it does not make your case easy. You seem to be suggesting that NO number, however imprecise, has any place here, and so you do not get to refute me by saying that I have to embrace arbitrary precision.
In any case, if you offer me some bets with more significant digits in the odds, my choices will reveal the cutoff to more significant digits. Wherever it may be, there will still be some bets I will and won’t take, and the number reflects that, which means it carries very real meaning.
Now, maybe I will hold the line at 54% exactly, not feeling any gain to thinking harder about the cutoff (as it gets harder AND less important to nail down further digits). Heck, maybe on some other issue I only care to go out to the nearest 10%. But so what? There are plenty of cases where I know my common sense belief probability to within 10%. That suggests such an estimate is not meaningless.
Again, meaningless is a very strong word, and it does not make your case easy.
To be precise, I wrote “meaningless, except perhaps as a vague figure of speech.” I agree that the claim would be too strong without that qualification, but I do believe that “vague figure of speech” is a fair summary of the meaningfulness that is to be found there. (Note also that the claim specifically applies to “common-sense conclusions and beliefs,” not things where there is a valid basis for employing mathematical models that yield numerical probabilities.)
In any case, if you offer me some bets with more significant digits in the odds, my choices will reveal the cutoff to more significant digits. Wherever it may be, there will still be some bets I will and won’t take, and the number reflects that, which means it carries very real meaning.
You seem to be saying that since you perceive this number as meaningful, you will be willing to act on it, and this by itself renders it meaningful, since it serves as guide for your actions. If we define “meaningful” to cover this case, then I agree with you, and this qualification should be added to my above statement. But the sense in which I used the term originally doesn’t cover this case.
Fair. Let me be precise too. I read your original statement as saying that numbers will never add meaning beyond what a vague figure of speech would, i.e. if you say “I strongly believe this” you cannot make your position more clear by attaching a number. That I disagree with. To me it seems clear that:
i) “Common-sense conclusions and beliefs” are held with varying levels of precision.
ii) Often even these beliefs are held with a level of precision that can be best described with a number. (Best=most succinctly, least misinterpretable, etc...indeed it seems to me that sometimes “best” could be replaced with “only.” You will never get people to understand 60% by saying “I reasonably strongly believe”...and yet your belief may be demonstrably closer to 60 than 50 or 70).
I don’t think your statement is defensible from a normal definition of “common sense conclusions,” but you may have internally defined it in such a way as to make your statement true, with a (I think) relatively narrow sense of “meaningfulness” also in mind. For instance if you ignore the role of numbers in transmission of belief from one party to the next, you are a big step closer to being correct.
I don’t think your statement is defensible from a normal definition of “common sense conclusions,” but you may have internally defined it in such a way as to make your statement true, with a (I think) relatively narrow sense of “meaningfulness” also in mind. For instance if you ignore the role of numbers in transmission of belief from one party to the next, you are a big step closer to being correct.
You have a very good point here. For example, a dialog like this could result in a real exchange of useful information:
A: “I think this project will probably fail.” B: “So, you mean you’re, like, 90% sure it will fail?” A: “Um… not really, more like 80%.”
I can imagine a genuine meeting of minds here, where B now has a very good idea of how confident A feels about his prediction. The numbers are still used as mere figures of speech, but “vague” is not a correct way to describe them, since the information has been transmitted in a more precise way than if A had just used verbal qualifiers.
So, I agree that “vague” should probably be removed from my original claim.
Therefore, there are only two ways in which you can arrive at a numerical probability estimate for a common-sense belief:
Translate your vague feeling of certainly into a number in some arbitrary manner. This however makes the number a mere figure of speech, which adds absolutely nothing over the usual human vague expressions for different levels of certainty.
Perform some probability calculation, which however has nothing to do with how your brain actually arrived at your common-sense conclusion, and then assign the probability number produced by the former to the latter. This is clearly fallacious.
On point #2, I agree with you. On point #1, I had the same reaction as xv15. Your example conversation is exactly how I would defend the use of numerical probabilities in conversation. I think you may have confused people with the phrase “vague figure of speech,” which was itself vague.
Vague relative to what? “No idea / kinda sure / pretty sure / very sure?”, the ways that people generally communicate about probability, are much worse. You can throw in other terms like “I suspect” and “absolutely certain” and “very very sure”, but it’s not even clear how these expressions of belief match up with others. In common speech, we really only have about 3-5 degrees of probability. That’s just not enough gradations.
In contrast, when expressing a percentage probability, people only tend to use multiples of 10, certain multiples of 5, 0.01%, 1%, 2%, 98%, 99% and 99.99%. If people use figures like 87%, or any decimal places other than the ones previously mentioned, it’s usually because they are deliberately being ridiculous. (And it’s no coincidence that your example uses multiples of 10.)
I agree with you that feelings of uncertainty are fuzzy, but they aren’t so fuzzy that we can get by with merely 3-5 gradations in all sorts of conversations. On some subjects, our communication becomes more precise when we have 10-20 gradations. Yet there are diminishing returns on more degrees of communicable certainty (due to reasons you correctly describe), so going any higher resolution than 10-20 degrees isn’t useful for anything except jokes.
The numbers are still used as mere figures of speech, but “vague” is not a correct way to describe them, since the information has been transmitted in a more precise way than if A had just used verbal qualifiers.
Yes. Gaining the 10-20 gradations that numbers allow when they are typically used does make conversations relatively more precise than just by tacking on “very very” to your statement of certainty.
It’s similar to the infamous 1-10 rating system for people’s attractiveness. Despite various reasons that rating people with numbers is distasteful, this ranking system persists because, in my view, people find it useful for communicating subjective assessments of attractiveness. Ugly-cute-hot is a 3-point scale. You could add in “gorgeous,” “beautiful,” or modifiers like “smoking hot,” but it’s unclear how these terms rank against each other (and they may express different types of attraction, rather than different degrees). Again, it’s hard to get more than 3-5 degrees using plain English. The 1-10 scale (with half-points, and 9.9) gives you about 20 gradations (though 1-3, and any half-point values below 5 are rarely used).
I think we have a generalized phenomenon where people resort to using numbers to describe their subjective feelings when common language doesn’t grant high enough resolution. 3-5 is good enough for some feelings (3 gives you negative, neutral, and positive for instance), but for some feelings we need more. Somewhere around 20 is the upper-bound of useful gradations.
I mostly agree with this assessment. However, the key point is that such uses of numbers should be seen as metaphorical. The literal meaning of a metaphor is typically nonsensical, but it works by somehow hacking the human understanding of language to successfully convey a point with greater precision than the most precise literal statement would allow, at least in as many words. (There are other functions of metaphors too, of course, but this one is relevant here.) And just like it is fallacious to understand a metaphor literally, it is similarly fallacious to interpret these numerical metaphors as useful for mathematical purposes. When it comes to subjective probabilities, however, I often see what looks like confusion on this point.
It is wrong to use a subjective probability that you got from someone else for mathematical purposes directly, for reasons I expand on in my comment here. But I don’t think that makes them metaphorical, unless you’re using a definition of metaphor that’s very different than the one I am. And you can use a subjective probability which you generated yourself, or combined with your own subjective probability, in calculations. Doing so just comes with the same caveats as using a probability from a study whose sample was too small, or which had some other bad but not entirely fatal flaw.
I will write a reply to that earlier comment of yours a bit later today when I’ll have more time. (I didn’t forget about it, it’s just that I usually answer lengthy comments that deserve a greater time investment later than those where I can fire off replies rapidly during short breaks.)
But in addition to the theme of that comment, I think you’re missing my point about the possible metaphorical quality of numbers. Human verbal expressions have their literal information content, but one can often exploit the idiosyncrasies of the human language interpretation circuits to effectively convey information altogether different from the literal meaning of one’s words. This gives rise to various metaphors and other figures of speech, which humans use in their communication frequently and effectively. (The process is more complex than this simple picture, since frequently used metaphors can eventually come to be understood as literal expressions of their common metaphorical meaning, and this process is gradual. There are also other important considerations about metaphors, but this simple observation is enough to support my point.)
Now, I propose that certain practical uses of numbers in communication should be seen that way too. A literal meaning of a number is that something can ultimately be counted, measured, or calculated to arrive at that number. A metaphorical use of a number, however, doesn’t convey any such meaning, but merely expects to elicit similar intuitive impressions, which would be difficult or even impossible to communicate precisely using ordinary words. And just like a verbal metaphor is nonsensical except for the non-literal intuitive point it conveys, and its literal meaning should be discarded, at least some practical uses of numbers in human conversations serve only to communicate intuitive points, and the actual values are otherwise nonsensical and should not be used for any other purposes—and even if they perhaps are, their metaphorical value should be clearly seen apart from their literal mathematical value.
Therefore, regardless of our disagreement about subjective probabilities (of which more in my planned reply), this is a separate important point I wanted to make.
okay. I still suspect I disagree with whatever you mean by mere “figures of speech,” but this rational truthseeker does not have infinite time or energy.
in any case, thank you for a productive and civil exchange.
Even if you believe that my position is fallacious, I am sure not the one to be accused of arbitrariness here. Arbitrariness is exactly what I object to, in the sense of insisting on the validity of numbers that lack both logically correct justification and clear error bars that would follow from it. And I’m asking the above question in full seriousness: a Bayesian probability calculation will give you as many significant digits as you want, so if you believe that it makes sense to extract a Bayesian probability with two significant digits from your common sense reasoning, why not more than that?
In any case, I have explained my position at length, and it would be nice if you addressed the substance of what I wrote instead of trying to come up with witty one-liner jabs. For those who want the latter, there are other places on the web full of people whose talent for such things is considerably greater than yours.
For those who want the latter, there are other places on the web full of people whose talent for such things is considerably greater than yours.
I specifically object to your implied argument in the grandparent. I will continue to reject comments that make that mistake regardless of how many times you insult me.
Look, in this thread, you have clearly been making jabs for rhetorical effect, without any attempt to argue in a clear and constructive manner. I am calling you out on that, and if you perceive that as insulting, then so be it.
Everything I wrote here has been perfectly honest and upfront, and written with the goal of eliciting rational counter-arguments from which I might perhaps change my opinion. I have neither the time nor the inclination for the sort of one-upmanship and showing off that you seem to be after, and even if I were, I would pursue it in some more suitable venue. (Where, among other things, one would indeed expect to see the sort of performance you’re striving for done in a much more skilled and entertaining way.)
Your map is not the territory. If you look a little closer you may find that my points are directed at the topic, and not your ego. In particular, take a second glance at this comment. The very example of betting illustrates the core problem with your position.
I am calling you out on that, and if you perceive that as insulting, then so be it.
The insult would be that you are telling me I’m bad at entertaining one-upmanship. I happen to believe I would be quite good at making such performances were I of a mind and in a context where it suited my goals (dealing with AMOGs, for example).
When dealing with intelligent agents, if you notice that what they are doing does not seem to be effective at achieving their goals it is time to notice your confusion. It is most likely that your model of their motives is inaccurate. Mind reading is hard.
Shultz does know nuthink. Slippery slopes do (arbitrarily) slide in both directions (to either Shultz to Omega in this case). Most importantly, if you cannot assign numbers to confidence levels you will lose money when you try to bet.
I tell you I believe X with 54% certainty. Who knows, that number could have been generated in a completely bogus way. But however I got here, this is where I am. There are bets about X that I will and won’t take, and guess what, that’s my cutoff probability right there. And by the way, now I have communicated to you where I am, in a way that does not further compound the error.
Meaningless is a very strong word.
In the face of such uncertainty, it could feel natural to take shelter in the idea of “inherent vagueness”...but this is reality, and we place our bets with real dollars and cents, and all the uncertainty in the world collapses to a number in the face of the expectation operator.
So why stop there? If you can justify 54%, then why not go further and calculate a dozen or two more significant digits, and stand behind them all with unshaken resolve?
You can, of course. For most situations, the effort is not worth the trade-off. But making a distinction between 1%, 25%, 50%. 75%. and 99% often is.
You can (at least formally) put error bars on the quantities that go into a Bayesian calculation. The problem, of course, is that error bars are short-hand for a distribution of possible values, and it’s not obvious what a distribution of probabilities means or should mean. Everything operational about probability functions is fully captured by their full set of expectation values, so this is no different than just immediately taking the mean, right?
Well, no. The uncertainties are a higher level model that not only makes predictions, but also calibrates how much these predictions are likely to move given new data.
It seems to me that this is somewhat related to the problem of logical uncertainty.
Again, meaningless is a very strong word, and it does not make your case easy. You seem to be suggesting that NO number, however imprecise, has any place here, and so you do not get to refute me by saying that I have to embrace arbitrary precision.
In any case, if you offer me some bets with more significant digits in the odds, my choices will reveal the cutoff to more significant digits. Wherever it may be, there will still be some bets I will and won’t take, and the number reflects that, which means it carries very real meaning.
Now, maybe I will hold the line at 54% exactly, not feeling any gain to thinking harder about the cutoff (as it gets harder AND less important to nail down further digits). Heck, maybe on some other issue I only care to go out to the nearest 10%. But so what? There are plenty of cases where I know my common sense belief probability to within 10%. That suggests such an estimate is not meaningless.
xv15:
To be precise, I wrote “meaningless, except perhaps as a vague figure of speech.” I agree that the claim would be too strong without that qualification, but I do believe that “vague figure of speech” is a fair summary of the meaningfulness that is to be found there. (Note also that the claim specifically applies to “common-sense conclusions and beliefs,” not things where there is a valid basis for employing mathematical models that yield numerical probabilities.)
You seem to be saying that since you perceive this number as meaningful, you will be willing to act on it, and this by itself renders it meaningful, since it serves as guide for your actions. If we define “meaningful” to cover this case, then I agree with you, and this qualification should be added to my above statement. But the sense in which I used the term originally doesn’t cover this case.
Fair. Let me be precise too. I read your original statement as saying that numbers will never add meaning beyond what a vague figure of speech would, i.e. if you say “I strongly believe this” you cannot make your position more clear by attaching a number. That I disagree with. To me it seems clear that:
i) “Common-sense conclusions and beliefs” are held with varying levels of precision. ii) Often even these beliefs are held with a level of precision that can be best described with a number. (Best=most succinctly, least misinterpretable, etc...indeed it seems to me that sometimes “best” could be replaced with “only.” You will never get people to understand 60% by saying “I reasonably strongly believe”...and yet your belief may be demonstrably closer to 60 than 50 or 70).
I don’t think your statement is defensible from a normal definition of “common sense conclusions,” but you may have internally defined it in such a way as to make your statement true, with a (I think) relatively narrow sense of “meaningfulness” also in mind. For instance if you ignore the role of numbers in transmission of belief from one party to the next, you are a big step closer to being correct.
xv15:
You have a very good point here. For example, a dialog like this could result in a real exchange of useful information:
A: “I think this project will probably fail.”
B: “So, you mean you’re, like, 90% sure it will fail?”
A: “Um… not really, more like 80%.”
I can imagine a genuine meeting of minds here, where B now has a very good idea of how confident A feels about his prediction. The numbers are still used as mere figures of speech, but “vague” is not a correct way to describe them, since the information has been transmitted in a more precise way than if A had just used verbal qualifiers.
So, I agree that “vague” should probably be removed from my original claim.
On point #2, I agree with you. On point #1, I had the same reaction as xv15. Your example conversation is exactly how I would defend the use of numerical probabilities in conversation. I think you may have confused people with the phrase “vague figure of speech,” which was itself vague.
Vague relative to what? “No idea / kinda sure / pretty sure / very sure?”, the ways that people generally communicate about probability, are much worse. You can throw in other terms like “I suspect” and “absolutely certain” and “very very sure”, but it’s not even clear how these expressions of belief match up with others. In common speech, we really only have about 3-5 degrees of probability. That’s just not enough gradations.
In contrast, when expressing a percentage probability, people only tend to use multiples of 10, certain multiples of 5, 0.01%, 1%, 2%, 98%, 99% and 99.99%. If people use figures like 87%, or any decimal places other than the ones previously mentioned, it’s usually because they are deliberately being ridiculous. (And it’s no coincidence that your example uses multiples of 10.)
I agree with you that feelings of uncertainty are fuzzy, but they aren’t so fuzzy that we can get by with merely 3-5 gradations in all sorts of conversations. On some subjects, our communication becomes more precise when we have 10-20 gradations. Yet there are diminishing returns on more degrees of communicable certainty (due to reasons you correctly describe), so going any higher resolution than 10-20 degrees isn’t useful for anything except jokes.
Yes. Gaining the 10-20 gradations that numbers allow when they are typically used does make conversations relatively more precise than just by tacking on “very very” to your statement of certainty.
It’s similar to the infamous 1-10 rating system for people’s attractiveness. Despite various reasons that rating people with numbers is distasteful, this ranking system persists because, in my view, people find it useful for communicating subjective assessments of attractiveness. Ugly-cute-hot is a 3-point scale. You could add in “gorgeous,” “beautiful,” or modifiers like “smoking hot,” but it’s unclear how these terms rank against each other (and they may express different types of attraction, rather than different degrees). Again, it’s hard to get more than 3-5 degrees using plain English. The 1-10 scale (with half-points, and 9.9) gives you about 20 gradations (though 1-3, and any half-point values below 5 are rarely used).
I think we have a generalized phenomenon where people resort to using numbers to describe their subjective feelings when common language doesn’t grant high enough resolution. 3-5 is good enough for some feelings (3 gives you negative, neutral, and positive for instance), but for some feelings we need more. Somewhere around 20 is the upper-bound of useful gradations.
I mostly agree with this assessment. However, the key point is that such uses of numbers should be seen as metaphorical. The literal meaning of a metaphor is typically nonsensical, but it works by somehow hacking the human understanding of language to successfully convey a point with greater precision than the most precise literal statement would allow, at least in as many words. (There are other functions of metaphors too, of course, but this one is relevant here.) And just like it is fallacious to understand a metaphor literally, it is similarly fallacious to interpret these numerical metaphors as useful for mathematical purposes. When it comes to subjective probabilities, however, I often see what looks like confusion on this point.
It is wrong to use a subjective probability that you got from someone else for mathematical purposes directly, for reasons I expand on in my comment here. But I don’t think that makes them metaphorical, unless you’re using a definition of metaphor that’s very different than the one I am. And you can use a subjective probability which you generated yourself, or combined with your own subjective probability, in calculations. Doing so just comes with the same caveats as using a probability from a study whose sample was too small, or which had some other bad but not entirely fatal flaw.
I will write a reply to that earlier comment of yours a bit later today when I’ll have more time. (I didn’t forget about it, it’s just that I usually answer lengthy comments that deserve a greater time investment later than those where I can fire off replies rapidly during short breaks.)
But in addition to the theme of that comment, I think you’re missing my point about the possible metaphorical quality of numbers. Human verbal expressions have their literal information content, but one can often exploit the idiosyncrasies of the human language interpretation circuits to effectively convey information altogether different from the literal meaning of one’s words. This gives rise to various metaphors and other figures of speech, which humans use in their communication frequently and effectively. (The process is more complex than this simple picture, since frequently used metaphors can eventually come to be understood as literal expressions of their common metaphorical meaning, and this process is gradual. There are also other important considerations about metaphors, but this simple observation is enough to support my point.)
Now, I propose that certain practical uses of numbers in communication should be seen that way too. A literal meaning of a number is that something can ultimately be counted, measured, or calculated to arrive at that number. A metaphorical use of a number, however, doesn’t convey any such meaning, but merely expects to elicit similar intuitive impressions, which would be difficult or even impossible to communicate precisely using ordinary words. And just like a verbal metaphor is nonsensical except for the non-literal intuitive point it conveys, and its literal meaning should be discarded, at least some practical uses of numbers in human conversations serve only to communicate intuitive points, and the actual values are otherwise nonsensical and should not be used for any other purposes—and even if they perhaps are, their metaphorical value should be clearly seen apart from their literal mathematical value.
Therefore, regardless of our disagreement about subjective probabilities (of which more in my planned reply), this is a separate important point I wanted to make.
okay. I still suspect I disagree with whatever you mean by mere “figures of speech,” but this rational truthseeker does not have infinite time or energy.
in any case, thank you for a productive and civil exchange.
Or, you could slide up your arbitrary and fallacious slippery slope and end up with Shultz.
Even if you believe that my position is fallacious, I am sure not the one to be accused of arbitrariness here. Arbitrariness is exactly what I object to, in the sense of insisting on the validity of numbers that lack both logically correct justification and clear error bars that would follow from it. And I’m asking the above question in full seriousness: a Bayesian probability calculation will give you as many significant digits as you want, so if you believe that it makes sense to extract a Bayesian probability with two significant digits from your common sense reasoning, why not more than that?
In any case, I have explained my position at length, and it would be nice if you addressed the substance of what I wrote instead of trying to come up with witty one-liner jabs. For those who want the latter, there are other places on the web full of people whose talent for such things is considerably greater than yours.
I specifically object to your implied argument in the grandparent. I will continue to reject comments that make that mistake regardless of how many times you insult me.
Look, in this thread, you have clearly been making jabs for rhetorical effect, without any attempt to argue in a clear and constructive manner. I am calling you out on that, and if you perceive that as insulting, then so be it.
Everything I wrote here has been perfectly honest and upfront, and written with the goal of eliciting rational counter-arguments from which I might perhaps change my opinion. I have neither the time nor the inclination for the sort of one-upmanship and showing off that you seem to be after, and even if I were, I would pursue it in some more suitable venue. (Where, among other things, one would indeed expect to see the sort of performance you’re striving for done in a much more skilled and entertaining way.)
Your map is not the territory. If you look a little closer you may find that my points are directed at the topic, and not your ego. In particular, take a second glance at this comment. The very example of betting illustrates the core problem with your position.
The insult would be that you are telling me I’m bad at entertaining one-upmanship. I happen to believe I would be quite good at making such performances were I of a mind and in a context where it suited my goals (dealing with AMOGs, for example).
When dealing with intelligent agents, if you notice that what they are doing does not seem to be effective at achieving their goals it is time to notice your confusion. It is most likely that your model of their motives is inaccurate. Mind reading is hard.
Shultz does know nuthink. Slippery slopes do (arbitrarily) slide in both directions (to either Shultz to Omega in this case). Most importantly, if you cannot assign numbers to confidence levels you will lose money when you try to bet.