I think mathematical formalism is a limited route. Mathematics gives you formal structures that might be useful for something, but demonstrating just what they are useful for is the real trick. . I’ve found that mathematicians often quite breezily assume their formalism applies to the world.
“Probability is defined as some formal structure.” Yawn. Until you show me that said mathematics actually solves the problems I’m trying to solve with concepts of probability, I’m uninterested.
we learn because probability theory is isomorphic to those mechanisms.
Showing how it is isomorphic is the real task.
This is fundamentally where Jaynes had it better than most. He starts by trying to solve the problem of which beliefs to have confidence and rely on. He’s solving a real problem, instead of creating mathematical objects and just applying real world labels to elements of them after the fact.
The easiest way to understand the philosophical dispute between the frequentist and the subjective Bayesian is to look at the classic biased coin:
…
I ask you. What is the difference between these two...
The difference between the two is that the subjective bayesian has a means to cope with the problem that at least most frequentists lack.
I mostly agree, finding why/how it is isomorphic is the important thing. But it is still isomorphic to more than one thing, frequency and subjective degree of belief included.
The two are still locked in a debate which is ultimately the result of interpreting one question in two different ways, and then answering the two seperate questions as if they were exclusive answers to one question. Exactly as the two argue about sound being there in the absence of observers.
The bayesian would give the same answer as the frequentist if he interpreted the question as the frequentist. Same goes for the sound realist, for the same reasons.
The two are still locked in a debate which is ultimately the result of interpreting one question in two different ways, and then answering the two seperate questions as if they were exclusive answers to one question. Exactly as the two argue about sound being there in the absence of observers.
Not exactly. The real argument is about what should be used for inference (both scientific and otherwise). The debate about “what probability actually is” is just another case of debating semantics as a proxy for debating what’s actually at stake.
Yes, I think we agree. Except that i don’t think that the fact that there is meaningful argument to be had about bayesian inference v.s. frequentist inference, means that the debate has not been centered around arguing about what probability is, which is a mistake; the same class of mistake as the mistake being made by the realists and idealists arguing over sound. The bayesian and the frequentist have proposed ways to settle their debate. And there are observations which act as evidence for bayesian inference, or frequentist inference. But exactly what experience should i expect if i think “probability is frequency” as opposed to if I think “probability is subjective degree of belief” ? Arguing about which inferences are optimal, is perfectly reasonable, but arguing about what thing probability really is, is silly.
Yes, I think we agree. Except that i don’t think that the fact that there is meaningful argument to be had about bayesian inference v.s. frequentist inference, means that the debate has not been centered around arguing about what probability is, which is a mistake...
Did I give the impression that I thought the argument about what probability is wasn’t a mistake?
How isormorphic it is remains to be seen. The infinite set digressions have not been particularly helpful to real problems.
The objective bayesian is free to estimate frequencies, and has done so, a la Jaynes. He explicitly identifies that both questions are answering different questions, and answers both.
I’m not aware of anyone doing this, but I think a frequentist could just as well interpret subjective degrees of belief in frequentist terms, but the sample space would be in informational terms, looking for transformation groups in states of knowledge.
“Probability” is a word used in the interpretation of probability theory.
Sometimes. I think if we’re trying to keep terms straight, you should separate probability_SubjectiveBayes, probability_Math, probability_Frequentist, and probability_HumanLanguage. You seem to conflate probability_Math and probability_HumanLanguage.
probability_SubjectiveBayes, probability_Math, probability_Frequentist, and probability_HumanLanguage. You seem to conflate probability_Math and probability_HumanLanguage.
probability\_SubjectiveBayes, probability\_Math, probability\_Frequentist, and probability\_HumanLanguage. You seem to conflate probability\_Math and probability\_HumanLanguage.
I think mathematical formalism is a limited route. Mathematics gives you formal structures that might be useful for something, but demonstrating just what they are useful for is the real trick. . I’ve found that mathematicians often quite breezily assume their formalism applies to the world.
“Probability is defined as some formal structure.” Yawn. Until you show me that said mathematics actually solves the problems I’m trying to solve with concepts of probability, I’m uninterested.
Showing how it is isomorphic is the real task.
This is fundamentally where Jaynes had it better than most. He starts by trying to solve the problem of which beliefs to have confidence and rely on. He’s solving a real problem, instead of creating mathematical objects and just applying real world labels to elements of them after the fact.
The difference between the two is that the subjective bayesian has a means to cope with the problem that at least most frequentists lack.
I mostly agree, finding why/how it is isomorphic is the important thing. But it is still isomorphic to more than one thing, frequency and subjective degree of belief included.
The two are still locked in a debate which is ultimately the result of interpreting one question in two different ways, and then answering the two seperate questions as if they were exclusive answers to one question. Exactly as the two argue about sound being there in the absence of observers.
The bayesian would give the same answer as the frequentist if he interpreted the question as the frequentist. Same goes for the sound realist, for the same reasons.
Not exactly. The real argument is about what should be used for inference (both scientific and otherwise). The debate about “what probability actually is” is just another case of debating semantics as a proxy for debating what’s actually at stake.
Quick edit: and your post helps make this clear.
Yes, I think we agree. Except that i don’t think that the fact that there is meaningful argument to be had about bayesian inference v.s. frequentist inference, means that the debate has not been centered around arguing about what probability is, which is a mistake; the same class of mistake as the mistake being made by the realists and idealists arguing over sound. The bayesian and the frequentist have proposed ways to settle their debate. And there are observations which act as evidence for bayesian inference, or frequentist inference. But exactly what experience should i expect if i think “probability is frequency” as opposed to if I think “probability is subjective degree of belief” ? Arguing about which inferences are optimal, is perfectly reasonable, but arguing about what thing probability really is, is silly.
Did I give the impression that I thought the argument about what probability is wasn’t a mistake?
I wasn’t sure.
Oh, well in rereading my comment I could see why it was ambiguous. Yeah, I think we agree.
How isormorphic it is remains to be seen. The infinite set digressions have not been particularly helpful to real problems.
The objective bayesian is free to estimate frequencies, and has done so, a la Jaynes. He explicitly identifies that both questions are answering different questions, and answers both.
I’m not aware of anyone doing this, but I think a frequentist could just as well interpret subjective degrees of belief in frequentist terms, but the sample space would be in informational terms, looking for transformation groups in states of knowledge.
Sometimes. I think if we’re trying to keep terms straight, you should separate probability_SubjectiveBayes, probability_Math, probability_Frequentist, and probability_HumanLanguage. You seem to conflate probability_Math and probability_HumanLanguage.
probability\_SubjectiveBayes, probability\_Math, probability\_Frequentist, and probability\_HumanLanguage. You seem to conflate probability\_Math and probability\_HumanLanguage.
Corrected. Thanks.