For questions of a continuous nature, you think that subjective probability is best expressed as a distribution over the continuous support, right? I view these sorts of distributions over distributions as that- there’s some continuous parameter potentially in the world (the proportion of white and black balls in the urn), and that continuous parameter may determine my subjective probability about binary events (whether ball #1001 is white or black).
Now, whether or not this formalism stretches to other ideas might be controversial. I might consider “the strength of the argument for Conclusion X” as having continuous support, possibly from 0 to 1, and so be able to express with my probability distribution over that how much more I expect to learn about the issue, but I can see reasons to avoid doing that.
[edit]That is, rather than modifying the likelihood ratios of all of the pieces of evidence for or against the argument being strong, I can modify my distribution on it. I think this runs in to trouble with, say, argument screening off authority- there’s a case where you really do want to modify the likelihood ratios.
“I view these sorts of distributions over distributions as that- there’s some continuous parameter potentially in the world (the proportion of white and black balls in the urn), and that continuous parameter may determine my subjective probability about binary events (whether ball #1001 is white or black).”
To me this just sounds like standard conditional probability. E.g. let p(x|I) be your subjective probability distribution over the parameter x (fraction of white balls in urn), given prior information I. Then
p(“ball 1001 is white”|I) = integral_x { p(“ball 1001 is white”|x,I)*p(x|I) } dx
So your belief in “ball 1001 is white” gets modulated by your belief distributions over x, sure. But I wouldn’t call this a “distribution over a distribution”. Yes, there is a set of likelihoods p(“ball 1001 is white”|x,I) which specify your subjective degree of belief in “ball 1001 is white” GIVEN various x, but in then end you want your degree of belief in “ball 1001 is white” considering ALL values that x might have and their relative plausibilities, i.e. you want the marginal likelihood to make your predictions.
(my marginalisation here ignores hypotheses outside the domain implied by there being a fraction of balls in the urn...)
For questions of a continuous nature, you think that subjective probability is best expressed as a distribution over the continuous support, right? I view these sorts of distributions over distributions as that- there’s some continuous parameter potentially in the world (the proportion of white and black balls in the urn), and that continuous parameter may determine my subjective probability about binary events (whether ball #1001 is white or black).
Now, whether or not this formalism stretches to other ideas might be controversial. I might consider “the strength of the argument for Conclusion X” as having continuous support, possibly from 0 to 1, and so be able to express with my probability distribution over that how much more I expect to learn about the issue, but I can see reasons to avoid doing that.
[edit]That is, rather than modifying the likelihood ratios of all of the pieces of evidence for or against the argument being strong, I can modify my distribution on it. I think this runs in to trouble with, say, argument screening off authority- there’s a case where you really do want to modify the likelihood ratios.
“I view these sorts of distributions over distributions as that- there’s some continuous parameter potentially in the world (the proportion of white and black balls in the urn), and that continuous parameter may determine my subjective probability about binary events (whether ball #1001 is white or black).”
To me this just sounds like standard conditional probability. E.g. let p(x|I) be your subjective probability distribution over the parameter x (fraction of white balls in urn), given prior information I. Then
p(“ball 1001 is white”|I) = integral_x { p(“ball 1001 is white”|x,I)*p(x|I) } dx
So your belief in “ball 1001 is white” gets modulated by your belief distributions over x, sure. But I wouldn’t call this a “distribution over a distribution”. Yes, there is a set of likelihoods p(“ball 1001 is white”|x,I) which specify your subjective degree of belief in “ball 1001 is white” GIVEN various x, but in then end you want your degree of belief in “ball 1001 is white” considering ALL values that x might have and their relative plausibilities, i.e. you want the marginal likelihood to make your predictions.
(my marginalisation here ignores hypotheses outside the domain implied by there being a fraction of balls in the urn...)