I think I agree completely with all of that. My earlier post was meant as an illustration that once you say C = A & B that you’re no longer dealing with a state of complete ignorance. You’re in complete ignorance of A and B, but not of C. In fact, C is completely defined as being the conjunction of A and B. I used the illustration of an envelope because as long as the envelope is closed you’re completely ignorant about its contents (by stipulation) but once you open it that’s no longer the case.
The answer for all three envelopes is, in the case of complete ignorance, 0.5.
So the probability that all three envelopes happen to contain a true hypothesis/proposition is 0.125 based on the assumption of independence. Since you said “mostly independent” does that mean you think we’re not allowed to assume complete independence? If the answer isn’t 0.125, what is it?
edit:
If your answer to the above is “still 0.5” then I have another scenario. You’re in total ignorance of A. B denotes the probability of rolling a a 6 on a regular die. What’s the probability that A & B are true? I’d say it has to be 1⁄12, even though it’s possible that A and B are not independent.
If you don’t know what A is and you don’t know what B is and C is the conjunction of A and B, then you don’t know what C is. This is precisely because, one cannot assume the independence of A and B. If you stipulate independence then you are no longer operating under conditions of complete ignorance. Strict, non-statistical independence can be represented as A!=B. A!=B tells you something about the hypothesis- its a fact about the hypothesis that we didn’t have in complete ignorance. This lets us give odds other than 1:1. See my comment here.
With regard to the scenario in the edit, the probability of A & B is 1⁄6 because we don’t know anything about independence. Now, you might say: “Jack, what are the chances A is dependent on B?! Surely most cases will involve A being something that has nothing to do with dice, much less something closely related to the throw of that particular dice.” But this kind of reasoning involves presuming things about the domain A purports to describe. The universe is really big and complex so we know there are lots of physical events A could conceivably describe. But what if the universe consisted only of one regular die that rolls once! If that is the only variable then A will =B. That we don’t live in such a universe or that this universe seems odd or unlikely are reasonable assumptions only because they’re based on our observations. But in the case of complete ignorance, by stipulation, we have no such observations. By definition, if you don’t know anything about A then you can’t know more about A&B then you know about B.
Complete ignorance just means 0.5, its just necessarily the case that when one specifies the hypothesis one provides analytic insight into the hypothesis which can easily change the probability. That is, any hypothesis that can be distinguished from an alternative hypothesis will give us grounds for ascribing a new probability to that hypothesis (based on the information used to distinguish it from alternative hypotheses).
Thanks for the explanation, that helped a lot. I expected you to answer 0.5 in the second scenario, and I thought your model was that total ignorance “contaminated” the model such that something + ignorance = ignorance. Now I see this is not what you meant. Instead it’s that something + ignorance = something. And then likewise something + ignorance + ignorance = something according to your model.
The problem with your model is that it clashes with my intuition (I can’t find fault with your arguments). I describe one such scenario here.
My intuition is that the probability of these two statements should not be the same:
A. “In order for us to succeed one of 12 things need to happen”
B. “In order for us to succeed all of these 12 things need to happen”
In one case we’re talking about a disjunction of 12 unknowns and in the second scenario we’re talking about a conjunction. Even if some of the “things” are not completely uncorrelated that shouldn’t affect the total estimate that much. My intuition is that saying P(A) = 1 − 0.5 ^ 12 and P(B) = 0.5 ^ 12. Worlds apart! As far as I can tell you would say that in both cases the best estimate we can make is 0.5. I introduce the assumption of independence (I don’t stipulate it) to fix this problem. Otherwise the math would lead me down a path that contradicts common sense.
I think I agree completely with all of that. My earlier post was meant as an illustration that once you say C = A & B that you’re no longer dealing with a state of complete ignorance. You’re in complete ignorance of A and B, but not of C. In fact, C is completely defined as being the conjunction of A and B. I used the illustration of an envelope because as long as the envelope is closed you’re completely ignorant about its contents (by stipulation) but once you open it that’s no longer the case.
So the probability that all three envelopes happen to contain a true hypothesis/proposition is 0.125 based on the assumption of independence. Since you said “mostly independent” does that mean you think we’re not allowed to assume complete independence? If the answer isn’t 0.125, what is it?
edit:
If your answer to the above is “still 0.5” then I have another scenario. You’re in total ignorance of A. B denotes the probability of rolling a a 6 on a regular die. What’s the probability that A & B are true? I’d say it has to be 1⁄12, even though it’s possible that A and B are not independent.
If you don’t know what A is and you don’t know what B is and C is the conjunction of A and B, then you don’t know what C is. This is precisely because, one cannot assume the independence of A and B. If you stipulate independence then you are no longer operating under conditions of complete ignorance. Strict, non-statistical independence can be represented as A!=B. A!=B tells you something about the hypothesis- its a fact about the hypothesis that we didn’t have in complete ignorance. This lets us give odds other than 1:1. See my comment here.
With regard to the scenario in the edit, the probability of A & B is 1⁄6 because we don’t know anything about independence. Now, you might say: “Jack, what are the chances A is dependent on B?! Surely most cases will involve A being something that has nothing to do with dice, much less something closely related to the throw of that particular dice.” But this kind of reasoning involves presuming things about the domain A purports to describe. The universe is really big and complex so we know there are lots of physical events A could conceivably describe. But what if the universe consisted only of one regular die that rolls once! If that is the only variable then A will =B. That we don’t live in such a universe or that this universe seems odd or unlikely are reasonable assumptions only because they’re based on our observations. But in the case of complete ignorance, by stipulation, we have no such observations. By definition, if you don’t know anything about A then you can’t know more about A&B then you know about B.
Complete ignorance just means 0.5, its just necessarily the case that when one specifies the hypothesis one provides analytic insight into the hypothesis which can easily change the probability. That is, any hypothesis that can be distinguished from an alternative hypothesis will give us grounds for ascribing a new probability to that hypothesis (based on the information used to distinguish it from alternative hypotheses).
Thanks for the explanation, that helped a lot. I expected you to answer 0.5 in the second scenario, and I thought your model was that total ignorance “contaminated” the model such that something + ignorance = ignorance. Now I see this is not what you meant. Instead it’s that something + ignorance = something. And then likewise something + ignorance + ignorance = something according to your model.
The problem with your model is that it clashes with my intuition (I can’t find fault with your arguments). I describe one such scenario here.
My intuition is that the probability of these two statements should not be the same:
A. “In order for us to succeed one of 12 things need to happen”
B. “In order for us to succeed all of these 12 things need to happen”
In one case we’re talking about a disjunction of 12 unknowns and in the second scenario we’re talking about a conjunction. Even if some of the “things” are not completely uncorrelated that shouldn’t affect the total estimate that much. My intuition is that saying P(A) = 1 − 0.5 ^ 12 and P(B) = 0.5 ^ 12. Worlds apart! As far as I can tell you would say that in both cases the best estimate we can make is 0.5. I introduce the assumption of independence (I don’t stipulate it) to fix this problem. Otherwise the math would lead me down a path that contradicts common sense.