In any real-life inference problem, nobody is going to tell you: “Here is the exact probability space with a precise, known probability for each outcome.” (I literally don’t know what such a thing would mean anyway). Is all inference thereby undefined? Like Einstein said, “As far as laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality”. If you can’t actually fulfill the axioms in real life, what’s the point?
If you still want to make inferences anyway, I think you’re going to have to adopt the Bayesian view. A probability distribution is never handed to us, and must always be extracted from what we know. And how you update your probabilities in response to new evidence also depends on what you know. If you can formalize exactly how, then you have a totally well-defined mathematical problem, hooray!
My point, then, is that we feel a problem isn’t well-defined exactly when we don’t know how to convert what we know into clear mathematics. (I’m really not trying to play a semantics game. This was an attempt to dissolve the concept of “well-defined” for probability questions.) But you can see a bit of a paradox when adding more information makes the mathematical problem harder, even though this shouldn’t make the problem any less “well-defined”.
Sure! The original problem invites assumptions of symmetry and integrality[1] and so on, and the implied assumptions result in a fully specified probability model. The adjusted problem adds information that necessarily breaks symmetry, and there is no correspondingly “obvious” set of assumptions that leads to a unique answer.
So yes, communicating probability problems requires some shared concepts and assumptions just like communication about anything else. I’ve previously commented here about my experience with word problems in teaching mathematics, and how the hardest part is not anything to do with the mathematics itself, but with background assumptions about interpreting the problem that were not explicitly taught.
I agree that there isn’t an “obvious” set of assumptions for the latter question that yields a unique answer. And granted I didn’t really dig into why entropy is a good measure, but I do think it ultimately yields the unique best guess given the information you have. The fact that it’s not obvious is rather the point! The question has a best answer, even if you don’t know what it is or how to give it.
In any real-life inference problem, nobody is going to tell you: “Here is the exact probability space with a precise, known probability for each outcome.” (I literally don’t know what such a thing would mean anyway). Is all inference thereby undefined? Like Einstein said, “As far as laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality”. If you can’t actually fulfill the axioms in real life, what’s the point?
If you still want to make inferences anyway, I think you’re going to have to adopt the Bayesian view. A probability distribution is never handed to us, and must always be extracted from what we know. And how you update your probabilities in response to new evidence also depends on what you know. If you can formalize exactly how, then you have a totally well-defined mathematical problem, hooray!
My point, then, is that we feel a problem isn’t well-defined exactly when we don’t know how to convert what we know into clear mathematics. (I’m really not trying to play a semantics game. This was an attempt to dissolve the concept of “well-defined” for probability questions.) But you can see a bit of a paradox when adding more information makes the mathematical problem harder, even though this shouldn’t make the problem any less “well-defined”.
Sure! The original problem invites assumptions of symmetry and integrality[1] and so on, and the implied assumptions result in a fully specified probability model. The adjusted problem adds information that necessarily breaks symmetry, and there is no correspondingly “obvious” set of assumptions that leads to a unique answer.
So yes, communicating probability problems requires some shared concepts and assumptions just like communication about anything else. I’ve previously commented here about my experience with word problems in teaching mathematics, and how the hardest part is not anything to do with the mathematics itself, but with background assumptions about interpreting the problem that were not explicitly taught.
The original problem never said that the random number generator only gives integers in the range 1 to 6. That’s an additional assumption.
I agree that there isn’t an “obvious” set of assumptions for the latter question that yields a unique answer. And granted I didn’t really dig into why entropy is a good measure, but I do think it ultimately yields the unique best guess given the information you have. The fact that it’s not obvious is rather the point! The question has a best answer, even if you don’t know what it is or how to give it.