Thanks a lot for the pointers! Greaves’ example seems to suffer the same problem, though, doesn’t it?
Suppose, for instance, you know only that I am about to draw a book from my shelf, and that each book on my shelf has a single-coloured cover. Then POI seems to suggest that you are rationally required to have credence ½ that it will be red (Q1=red, Q2 = not-red; and you have no evidence bearing on whether or not the book is red), but also that you are rationally required to have credence 1/n that it will be red, where n is the ‘number of possible colours’ (Qi = ith colour; and you have no evidence bearing on what colour the book is).)
We have information about the set and distribution of colors, and assigning 50% credence to the color red does not use that information.
The cube factory problem does suffer less from this, cool!
A factory produces cubes with side-length between 0 and 1 foot; what is the probability that a randomly chosen cube has side-length between 0 and 1⁄2 a foot? The classical intepretation’s answer is apparently 1⁄2, as we imagine a process of production that is uniformly distributed over side-length. But the question could have been given an equivalent restatement: A factory produces cubes with face-area between 0 and 1 square-feet; what is the probability that a randomly chosen cube has face-area between 0 and 1⁄4 square-feet? Now the answer is apparently 1⁄4, as we imagine a process of production that is uniformly distributed over face-area.
I wonder if one should simply model this hierarchically, assigning equal credence to the idea that the relevant measure in cube production is side length or volume. For example, we might have information about cube bottle customers that want to fill their cubes with water. Because the customers vary in how much water they want to fit in their cube bottles, it seems to me that we should put more credence into partitioning it according to volume. Or if we’d have some information that people often want to glue the cubes under their shoes to appear taller, the relevant measure would be the side length. Currently, we have no information like this, so we should assign equal credence to both measures.
I don’t think Greaves’ example suffers the same problem actually—if we truly don’t know anything about what the possible colours are (just that each book has one colour), then there’s no reason to prefer {red, yellow, blue, other} over {red, yellow, blue, green, other}.
In the case of truly having no information, I think it makes sense to use Jeffreys prior in the box factory case because that’s invariant to reparametrisation, so it doesn’t matter whether the problem is framed in terms of length, area, volume, or some other parameterisation. I’m not sure what that actually looks like in this case though
Hm, but if we don’t know anything about the possible colours, the natural prior to assume seems to me to give all colors the same likelihood. It seems arbitrary to decide to group a subsection of colors under the label “other”, and pretend like it should be treated like a hypothesis on equal footing with the others in your given set, which are single colors.
Thanks a lot for the pointers! Greaves’ example seems to suffer the same problem, though, doesn’t it?
We have information about the set and distribution of colors, and assigning 50% credence to the color red does not use that information.
The cube factory problem does suffer less from this, cool!
I wonder if one should simply model this hierarchically, assigning equal credence to the idea that the relevant measure in cube production is side length or volume. For example, we might have information about cube bottle customers that want to fill their cubes with water. Because the customers vary in how much water they want to fit in their cube bottles, it seems to me that we should put more credence into partitioning it according to volume. Or if we’d have some information that people often want to glue the cubes under their shoes to appear taller, the relevant measure would be the side length. Currently, we have no information like this, so we should assign equal credence to both measures.
I don’t think Greaves’ example suffers the same problem actually—if we truly don’t know anything about what the possible colours are (just that each book has one colour), then there’s no reason to prefer {red, yellow, blue, other} over {red, yellow, blue, green, other}.
In the case of truly having no information, I think it makes sense to use Jeffreys prior in the box factory case because that’s invariant to reparametrisation, so it doesn’t matter whether the problem is framed in terms of length, area, volume, or some other parameterisation. I’m not sure what that actually looks like in this case though
Hm, but if we don’t know anything about the possible colours, the natural prior to assume seems to me to give all colors the same likelihood. It seems arbitrary to decide to group a subsection of colors under the label “other”, and pretend like it should be treated like a hypothesis on equal footing with the others in your given set, which are single colors.
Yeah, Jeffreys prior seems to make sense here.