“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...)
“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...)