I've been following, but I'm still nonplussed as to your use of the type-token distinction in this context. The comment of mine which was the parent for your type-token observation had a specific request: show me the specific mistake in my math, rather than appeal to a verbal presentation of a non-formal, intuitive explanation.

Take a bag with 1 red marble and 9 green marbles. There is a type "green marble" and it has 9 tokens. The experiences of drawing any particular green marble, while token-distinct are type-identical. It seems that what matters when we compute our credence for the proposition "the next marble I draw will be green" is the tokens, not the types. When you formalize the bag problem accordingly, probability theory gives you answers that seem quite robust from a math point of view.

If you start out ignorant of how many marbles the bag has of each color, you can ask questions like "given that I just took two green marbles in a row, what is my credence in the proposition 'the next marble I draw will be green'". You can compute things like the expected number of green marbles left in the bag. In the bag problem, IOW, we are quantifying our uncertainty over tokens, while taking types to be a fixed feature of the situation. (Which of course is only a convention of this kind of exercise: with precise enough instruments we could distinguish all ten individual marbles.)

Statements like "information is gained" or "information is lost" are vague and imprecise, with the consequence that a motivated interpretation of the problem statement will support whichever statement we happen to favor. The point of formalizing probability is precisely that we get to replace such vague statements with precisely quantifiable formalizations, which leave no wiggle room for interpretation.

If you have a formalism which shows, in that manner, why the answer to the Sleeping Beauty question is 1/2, I would love to see it: I have no attachment any longer to "my opinion" on the topic.

My questions to you, then, are: a) given your reasons for "worrying about types rather than tokens" in this situation, how do you formally quantify your uncertainty over various propositions, as I do in the spreadsheet I've linked to earlier? b) what justifies "worrying about types rather than tokens" in this situation, where every other discussion of probability "worries about tokens" in the sense I've outlined above in reference to the bag of marbles? c) how do you apply the type-token distinction in other problems, say, in the case of the Tuesday Boy?

"No knowledge has been lost"?!?

Memories are knowledge - they are knowledge about past perceptions. They have been lost - because they have been chemically deleted by the amnesia-inducing drug. If they had not been lost, Beauty's probability estimates would be very different at each interview - so evidently the lost information was important in influencing Beauty's probability estimates.

That should be all you need to know to establish that the deletion of Beauty's memories changes her priors, and thereby alters her subjective probability estimates. Beauty awakens, not knowing if she has previously been interviewed - because of her memory loss. She knew whether she had previously been interviewed at the start of the experiment - she hadn't. So: that illustrates which memories have been deleted, and why her uncertainty has increased.