The idea that “probability” is some preexisting thing that needs to be “interpreted” as something always seemed a little bit backwards to me. Isn’t it more straightforward to say:
Beliefs exist, and obey the Kolmogorov axioms (at least, “correct” beliefs do, as formalized by generalizations of logic (Cox’s theorem), or by possible-world-counting). This is what we refer to as “bayesian probabilities”, and code into AIs when we want to them to represent beliefs.
Measures over imaginary event classes / ensembles also obey the Kolmogorov axioms. “Frequentist probabilities” fall into this category.
Personally I mostly think about #1 because I’m interested in figuring out what I should believe, not about frequencies in arbitrary ensembles. But the fact is that both of these obey the same “probability” axioms, the Kolmogorov axioms. Denying one or the other because “probability” must be “interpreted” as exclusively either #1 or #2 is simply wrong (but that’s what frequentists effectively do when they loudly shout that you “can’t” apply probability to beliefs).
Now, sometimes you do need to interpret “probability” as something—in the specific case where someone else makes an utterance containing the word “probability” and you want to figure out what they meant. But the answer there is probably that in many cases people don’t even distinguish between #1 and #2, because they’ll only commit to a specific number when there’s a convenient instance of #2 that make #1 easy to calculate. For instance, saying 1⁄6 for a roll of a “fair” die.
People often act as though their utterances about probability refer to #1 though. For instance when they misinterpret p-values as the post-data probability of the null hypothesis and go around believing that the effect is real...
You might be interested in some work by Glenn Shafer and Vladimir Vovk about replacing measure theory with a game-theoretic approach. They have a website here, and I wrote a lay review of their first book on the subject here.
I have also just now discovered that a new book is due out in May, which presumably captures the last 18 years or so of research on the subject.
This isn’t really a direct response to your post, except insofar as I feel broadly the same way about the Kolmogorov axioms as you do about interpreting their application to phenomena, and this is another way of getting at the same intuitions.
I clicked this because it seemed interesting, but reading the Q&A:
In atypical game we consider, one player offers bets, another decides how to bet, and a third decides the outcome of the bet. We often call the first player Forecaster, the second Skeptic, and the third Reality.
How is this any different from the classical Dutch Book argument, that unless you maintain beliefs as probabilities you will inevitably lose money?
It’s just a different way of arriving at the same conclusions. The whole project is developing game-theoretic proofs for results in probability and finance.
The pitch is, rather than using a Dutch Book argument as a separate singular argument, they make those intuitions central as a mechanism of proof for all of probability (or at least the core of it, thus far).
The idea that “probability” is some preexisting thing that needs to be “interpreted” as something always seemed a little bit backwards to me. Isn’t it more straightforward to say:
Beliefs exist, and obey the Kolmogorov axioms (at least, “correct” beliefs do, as formalized by generalizations of logic (Cox’s theorem), or by possible-world-counting). This is what we refer to as “bayesian probabilities”, and code into AIs when we want to them to represent beliefs.
Measures over imaginary event classes / ensembles also obey the Kolmogorov axioms. “Frequentist probabilities” fall into this category.
Personally I mostly think about #1 because I’m interested in figuring out what I should believe, not about frequencies in arbitrary ensembles. But the fact is that both of these obey the same “probability” axioms, the Kolmogorov axioms. Denying one or the other because “probability” must be “interpreted” as exclusively either #1 or #2 is simply wrong (but that’s what frequentists effectively do when they loudly shout that you “can’t” apply probability to beliefs).
Now, sometimes you do need to interpret “probability” as something—in the specific case where someone else makes an utterance containing the word “probability” and you want to figure out what they meant. But the answer there is probably that in many cases people don’t even distinguish between #1 and #2, because they’ll only commit to a specific number when there’s a convenient instance of #2 that make #1 easy to calculate. For instance, saying 1⁄6 for a roll of a “fair” die.
People often act as though their utterances about probability refer to #1 though. For instance when they misinterpret p-values as the post-data probability of the null hypothesis and go around believing that the effect is real...
You might be interested in some work by Glenn Shafer and Vladimir Vovk about replacing measure theory with a game-theoretic approach. They have a website here, and I wrote a lay review of their first book on the subject here.
I have also just now discovered that a new book is due out in May, which presumably captures the last 18 years or so of research on the subject.
This isn’t really a direct response to your post, except insofar as I feel broadly the same way about the Kolmogorov axioms as you do about interpreting their application to phenomena, and this is another way of getting at the same intuitions.
There’s a Q&A with one of the authors here which explains a little about the purpose of the approach, mainly talks about the new book.
I clicked this because it seemed interesting, but reading the Q&A:
How is this any different from the classical Dutch Book argument, that unless you maintain beliefs as probabilities you will inevitably lose money?
It’s just a different way of arriving at the same conclusions. The whole project is developing game-theoretic proofs for results in probability and finance.
The pitch is, rather than using a Dutch Book argument as a separate singular argument, they make those intuitions central as a mechanism of proof for all of probability (or at least the core of it, thus far).