So in “formalism”, I understand that we are to say: “probability models frequency”, “probability models subjective degrees of belief” and “probability is the set of mathematical discoveries we have made, which deal with [ ], including such things as Bayes’s theorem”.
Whereas at the moment, Bayesians say: “probability is a measure of subjective degrees of belief”, “probability isn’t frequency”, and “probability theory is the set of mathematical discoveries we have made, which deal with probability, including such things as Bayes’s theorem”.
And frequentists say: “probability is long-run frequency”, “probability isn’t subjective degrees of belief”, and “probability theory is the set of mathematical discoveries we have made, which deal with probability, including such things as Bayes’s theorem”.
I like the Bayesian version. But the frequentist version doesn’t confuse me; I understand perfectly well that these are merely competing interpretations, and I’ve never felt the urge to argue specifically about whether probability is degrees of belief or is frequency—nor have I ever seen anyone else do so. Clearly that would be a stupid argument, just like the definitional dispute about sound. However, sensible people do use these terms, arguing about whether probability ‘is’ one or the other, as a proxy for a more substantive argument about which is the better—i.e. more philosophically parsimonious, and having better practical outcomes—interpretation. (Actually they are more likely to phrase the argument as “probability should be considered to be X”, and then say probability is X when they aren’t having the argument, but hey.)
As for the “formalist” version, firstly it puts the frequentist and Bayesian interpretations on a level footing. Even if sensible people were wasting time and effort arguing specifically over a mere definition, the cost of conceding ground to the problematic frequentist interpretation outweighs any benefit from ending that argument, in comparison to the option of simply carrying on using the language of the Bayesian.
Furthermore it appears to me that probability theory, given this use of language, lacks a referent. Probability theory has been renamed (simply) probability, and it no longer appears to be the theory of anything. Whether or not this use of language could be considered wrong per se, it hardly seems to be clearing up any philosophical confusion! If I ask “what is this thing that I am computing using Bayes’s theorem?”, the answer is no longer “the posterior probability”—if probability is the new word for the mathematical tools of probability theory, the phrase posterior probability no longer means anything. So perhaps I’ll have to invent a new word to refer to the same thing that the word probability used to refer to.
Do you begin to see why I think this is a waste of time?
NB: I think we’re making much more progress than I made with user:potato. That’s what I mean about the difficulty of having to argue with someone who is inarticulate, i.e. can’t state his case properly.
“probability is the set of mathematical discoveries we have made, which deal with [ ], including such things as Bayes’s theorem”.
Probably better put in terms of being a formal system, rather than “a set of mathematical discoveries”. But I fear that tends towards begging the question!
As for the “formalist” version, firstly it puts the frequentist and Bayesian interpretations on a level footing. Even if sensible people were wasting time and effort arguing specifically over a mere definition, the cost of conceding ground to the problematic frequentist interpretation outweighs any benefit from ending that argument, in comparison to the option of simply carrying on using the language of the Bayesian.
This treatment (notably the use of terms like “conceding ground”) suggests that you are engaging in a “political”/”debate” mode rather than a “truth-seeking” mode. This leads me to believe that we have more to lose by accepting the “Bayesian/Frequentist” duality than by dissolving it entirely and changing our terminology to match. This matches my impression of previous forays into the “Bayesian/Frequentist” ‘holy wars’.
If politics is mind-killing, then it must certainly be avoided even at great cost with respect to our most basic tools of rationality.
Do you begin to see why I think this is a waste of time?
Indeed, though in that case you’ve spent far more time on this than most who exercised the default ‘ignore’ option.
If I ask “what is this thing that I am computing using Bayes’s theorem?”, the answer is no longer “the posterior probability”—if probability is the new word for the mathematical tools of probability theory, the phrase posterior probability no longer means anything. So perhaps I’ll have to invent a new word to refer to the same thing that the word probability used to refer to.
A good point.
That’s what I mean about the difficulty of having to argue with someone who is inarticulate, i.e. can’t state his case properly.
I understood what you meant—I just did not see any inarticulateness on the part of User:potato.
I’ve never felt the urge to argue specifically about whether probability is degrees of belief or is frequency—nor have I ever seen anyone else do so.
I normally see this being explicitly the subject on Bayesian/Frequentist debates, and many long conversations with philosophers have revolved around whether “equating probability with subjective belief” is an “ontological confusion”.
This treatment (notably the use of terms like “conceding ground”) suggests that you are engaging in a “political”/”debate” mode rather than a “truth-seeking” mode.
Duly noted. I’ll try not to give this impression in future.
I normally see this being explicitly the subject on Bayesian/Frequentist debates, and many long conversations with philosophers have revolved around whether “equating probability with subjective belief” is an “ontological confusion”.
I may have simply failed to notice these arguments taking place. In order to dissolve any such ostensible ontological question, I’d recommend pointing out that to say probability is one or other thing is merely a statement to the effect that one interpretation is preferred for some reason by the writer—since both interpretations satisfy the Cox postulates or Kolmogorov axioms, we could define probability to be either subjective degrees of belief or long-run frequency, and make sound and rational inferences in either case (albeit perhaps not with the same efficiency). This should be enough to persuade an otherwise sensible person that he’s engaged in a futile argument about definitions.
Formalism attempts to solve the problem by effectively tabooing the concept of probability such that it no longer has a definition. Although we might be able to get around the problem that I mentioned by answering the question “”what is this thing that I am computing using Bayes’s theorem?” by saying “the posterior subjective degree of belief” or “the posterior frequency”, it’s easy to see how the same kind of philosophers would end up arguing over whether, in the case of a coin flip for example, we are really talking about prior and posterior subjective degrees of belief, or about prior and posterior long-run frequencies. And we would have lost the use of the word “probability”, which makes our messages shorter than they would otherwise be.
To the extent that there is such a thing as the proper use of words, to delete useful words from our vocabulary in order to (probably unsuccessfully) prevent people from having a definitional argument that could best be dispelled by introducing them to such notions as “dissolving the question” and reductionism isn’t it. On the other hand I’ll give user:potato credit for exposing an issue that may be more problematic than I at first believed.
I expect that we are substantially in agreement at this point.
So in “formalism”, I understand that we are to say: “probability models frequency”, “probability models subjective degrees of belief” and “probability is the set of mathematical discoveries we have made, which deal with [ ], including such things as Bayes’s theorem”.
Whereas at the moment, Bayesians say: “probability is a measure of subjective degrees of belief”, “probability isn’t frequency”, and “probability theory is the set of mathematical discoveries we have made, which deal with probability, including such things as Bayes’s theorem”.
And frequentists say: “probability is long-run frequency”, “probability isn’t subjective degrees of belief”, and “probability theory is the set of mathematical discoveries we have made, which deal with probability, including such things as Bayes’s theorem”.
I like the Bayesian version. But the frequentist version doesn’t confuse me; I understand perfectly well that these are merely competing interpretations, and I’ve never felt the urge to argue specifically about whether probability is degrees of belief or is frequency—nor have I ever seen anyone else do so. Clearly that would be a stupid argument, just like the definitional dispute about sound. However, sensible people do use these terms, arguing about whether probability ‘is’ one or the other, as a proxy for a more substantive argument about which is the better—i.e. more philosophically parsimonious, and having better practical outcomes—interpretation. (Actually they are more likely to phrase the argument as “probability should be considered to be X”, and then say probability is X when they aren’t having the argument, but hey.)
As for the “formalist” version, firstly it puts the frequentist and Bayesian interpretations on a level footing. Even if sensible people were wasting time and effort arguing specifically over a mere definition, the cost of conceding ground to the problematic frequentist interpretation outweighs any benefit from ending that argument, in comparison to the option of simply carrying on using the language of the Bayesian.
Furthermore it appears to me that probability theory, given this use of language, lacks a referent. Probability theory has been renamed (simply) probability, and it no longer appears to be the theory of anything. Whether or not this use of language could be considered wrong per se, it hardly seems to be clearing up any philosophical confusion! If I ask “what is this thing that I am computing using Bayes’s theorem?”, the answer is no longer “the posterior probability”—if probability is the new word for the mathematical tools of probability theory, the phrase posterior probability no longer means anything. So perhaps I’ll have to invent a new word to refer to the same thing that the word probability used to refer to.
Do you begin to see why I think this is a waste of time?
NB: I think we’re making much more progress than I made with user:potato. That’s what I mean about the difficulty of having to argue with someone who is inarticulate, i.e. can’t state his case properly.
Probably better put in terms of being a formal system, rather than “a set of mathematical discoveries”. But I fear that tends towards begging the question!
This treatment (notably the use of terms like “conceding ground”) suggests that you are engaging in a “political”/”debate” mode rather than a “truth-seeking” mode. This leads me to believe that we have more to lose by accepting the “Bayesian/Frequentist” duality than by dissolving it entirely and changing our terminology to match. This matches my impression of previous forays into the “Bayesian/Frequentist” ‘holy wars’.
If politics is mind-killing, then it must certainly be avoided even at great cost with respect to our most basic tools of rationality.
Indeed, though in that case you’ve spent far more time on this than most who exercised the default ‘ignore’ option.
A good point.
I understood what you meant—I just did not see any inarticulateness on the part of User:potato.
I normally see this being explicitly the subject on Bayesian/Frequentist debates, and many long conversations with philosophers have revolved around whether “equating probability with subjective belief” is an “ontological confusion”.
Duly noted. I’ll try not to give this impression in future.
I may have simply failed to notice these arguments taking place. In order to dissolve any such ostensible ontological question, I’d recommend pointing out that to say probability is one or other thing is merely a statement to the effect that one interpretation is preferred for some reason by the writer—since both interpretations satisfy the Cox postulates or Kolmogorov axioms, we could define probability to be either subjective degrees of belief or long-run frequency, and make sound and rational inferences in either case (albeit perhaps not with the same efficiency). This should be enough to persuade an otherwise sensible person that he’s engaged in a futile argument about definitions.
Formalism attempts to solve the problem by effectively tabooing the concept of probability such that it no longer has a definition. Although we might be able to get around the problem that I mentioned by answering the question “”what is this thing that I am computing using Bayes’s theorem?” by saying “the posterior subjective degree of belief” or “the posterior frequency”, it’s easy to see how the same kind of philosophers would end up arguing over whether, in the case of a coin flip for example, we are really talking about prior and posterior subjective degrees of belief, or about prior and posterior long-run frequencies. And we would have lost the use of the word “probability”, which makes our messages shorter than they would otherwise be.
To the extent that there is such a thing as the proper use of words, to delete useful words from our vocabulary in order to (probably unsuccessfully) prevent people from having a definitional argument that could best be dispelled by introducing them to such notions as “dissolving the question” and reductionism isn’t it. On the other hand I’ll give user:potato credit for exposing an issue that may be more problematic than I at first believed.
I expect that we are substantially in agreement at this point.