Well, it's less of a conundrum in the case of the die because it's easier to estimate the priors. That's the power and difficulty of using Bayesian reasoning; you need some estimate of how likely something is to be true before you perform the statistical test.
In case of the six-sided die, we have a good physical model and years of memory to know that there should be a 1/6 chance of any number coming up, so it's easy to estimate that prior. Similarly, when doing a disease screening test, we have data on how common diseases are in the general population that can be easily used.
The problem comes when using Bayesian methods on unsolved scientific problems. Suppose you don't know if the Earth goes around the Sun or the other way around, it's the early Renaissance and you've gathered some data that could indicate a probability of one or the other. What's the prior odds that one theory is correct? You have no idea, that's why you're investigating! The worry when using Bayes's theorem to replace deductive p-value methods is that the prior probabilities may just be made up out of baseless intuition, and skew the final calculation. (It can still be used effectively, you just have to get a little fancy.)
In case of the six-sided die, we have a good physical model and years of memory to know that there should be a 1/6 chance of any number coming up, so it's easy to estimate that prior. Similarly, when doing a disease screening test, we have data on how common diseases are in the general population that can be easily used.
The problem comes when using Bayesian methods on unsolved scientific problems. Suppose you don't know if the Earth goes around the Sun or the other way around, it's the early Renaissance and you've gathered some data that could indicate a probability of one or the other. What's the prior odds that one theory is correct? You have no idea, that's why you're investigating! The worry when using Bayes's theorem to replace deductive p-value methods is that the prior probabilities may just be made up out of baseless intuition, and skew the final calculation. (It can still be used effectively, you just have to get a little fancy.)