Could how you update your priors be dependent on what concepts you choose to represent the situation with?
I mean, suppose the parent says “I have two children, at least one of whom is a boy. So, I have a boy and another child whose gender I’m not mentioning”. It seems like that second sentence doesn’t add any new information- it parses to me like just a rephrasing of the first sentence. But now you’ve been presented with two seemingly incompatible ways of conceptualizing the scenario- either as two children of unknown gender, of whom one is a boy (suggesting a 2⁄3 chance of both being boys), or as one boy and one child of unknown gender (suggesting a 1⁄2 chance of both being boys). Having been prompted which both models, which should you choose?
It seems like one ought to have more predictive power than the other, and therefore ought to be chosen regardless of exactly how the parent phrases the statement. But it’s hard to think of a way to determine which would be more predictive in practice. If I were to select all of the pairs of two siblings in the world, discard the pairs of sisters, choose one at random and ask you to bet on whether they were both boys, you’d be wise to bet at 2⁄3 odds. But if I were to select all of the brothers with one sibling in the world and choose one along with their sibling at random, you’d want to bet at 1⁄2 odds. In the scenario above, are the unknown factors determining whether both children are boys more like that first randomization process, or more like the second? Or, maybe we have so little information about the process generating the statement that we really have no basis for deciding which is more predictive, and should just choose the simpler model?
Could how you update your priors be dependent on what concepts you choose to represent the situation with?
I mean, suppose the parent says “I have two children, at least one of whom is a boy. So, I have a boy and another child whose gender I’m not mentioning”. It seems like that second sentence doesn’t add any new information- it parses to me like just a rephrasing of the first sentence. But now you’ve been presented with two seemingly incompatible ways of conceptualizing the scenario- either as two children of unknown gender, of whom one is a boy (suggesting a 2⁄3 chance of both being boys), or as one boy and one child of unknown gender (suggesting a 1⁄2 chance of both being boys). Having been prompted which both models, which should you choose?
It seems like one ought to have more predictive power than the other, and therefore ought to be chosen regardless of exactly how the parent phrases the statement. But it’s hard to think of a way to determine which would be more predictive in practice. If I were to select all of the pairs of two siblings in the world, discard the pairs of sisters, choose one at random and ask you to bet on whether they were both boys, you’d be wise to bet at 2⁄3 odds. But if I were to select all of the brothers with one sibling in the world and choose one along with their sibling at random, you’d want to bet at 1⁄2 odds. In the scenario above, are the unknown factors determining whether both children are boys more like that first randomization process, or more like the second? Or, maybe we have so little information about the process generating the statement that we really have no basis for deciding which is more predictive, and should just choose the simpler model?