I don’t see how this helps. You can have a 1:1 prior over the question you’re interested in (like U1), however, to compute the likelihood ratios, it seems you would need a joint prior over everything of interest (including LL and E). There are specific cases where you can get a likelihood ratio without a joint prior (such as, likelihood of seeing some coin flips conditional on coin biases) but this doesn’t seem like a case where this is feasible.
To be clear, this is an equivalent way of looking at normal prior-ful inference, and doesn’t actually solve any practical problem you might have. I mostly see it as a demonstration of how you can shove everything into stuff that gets expressed as likelihood functions.
I don’t see how this helps. You can have a 1:1 prior over the question you’re interested in (like U1), however, to compute the likelihood ratios, it seems you would need a joint prior over everything of interest (including LL and E). There are specific cases where you can get a likelihood ratio without a joint prior (such as, likelihood of seeing some coin flips conditional on coin biases) but this doesn’t seem like a case where this is feasible.
To be clear, this is an equivalent way of looking at normal prior-ful inference, and doesn’t actually solve any practical problem you might have. I mostly see it as a demonstration of how you can shove everything into stuff that gets expressed as likelihood functions.