A very plausible normative principle relating subjective degree of belief to objective chance is David Lewis’ “Principal Principle”. In a simplified version, this principle says that if you know the objective chance of some inherently chancy outcome, then your degree of belief in that outcome should equal the chance. Thus, if you know that the coin is fair, then you should have degree of belief 1/2 that it will come up heads.
This has some added bite because the chance information overrules a lot of other information – if you know the coin is fair, then it doesn’t matter how it happened to come up on the last 1000 flips, you should still believe in heads to degree 1/2. Even if the last 1000 flips were all tails – this is one idea of what’s fallacious about the gambler’s fallacy (or inverse gambler’s fallacy).
Of course, some sorts of information can overrule the chance information – if a very accurate fortuneteller has told you that the coin will come up heads, then maybe you should believe to a degree higher than 1/2, even though you still believe the coin is fair. This sort of information is what Lewis called “inadmissible” information. The question for the Principal Principle then is just what counts as inadmissible information?
To answer this, I think we need to consider just what chance really is. On one notion of chance, it requires that the world be objectively indeterministic, so that there is no fact of the matter about future chancy events. On this account, the idea of an accurate fortuneteller for chancy events doesn’t even make sense. This might be a natural view of chance that arises from the many-worlds interpretation of quantum mechanics. On this view, the chance of an event could potentially depend on anything for which there is a fact of the matter – but this only includes facts about the past and present. But since you’d need to know all this information (or the relevant parts anyway) to know the chances, there will trivially be no possibility of inadmissible evidence, so the Principal Principle stands (if at all) in a very simple form!
But there are other notions of chance I’ve heard people talk about. One is supposed to be compatible with strict determinism. I don’t know too many of the details, but I suspect that the idea is that there’s some natural class of “nearby worlds”, and chance is just some sort of probability measure on those worlds. This can definitely give rise to non-extreme values for chances, even though there is no possibility other than necessity. However, on this interpretation of chance, I don’t see why anything like the Principal Principle would have any normative force at all. I suppose if you can somehow narrow things down enough to know what the chances are, but can’t eliminate any of the worlds in the class that defines the chances, then it would make sense. But it’s far from clear to me why this situation would be at all common.
Then of course there’s Lewis’ own characterization of chance. I believe his idea is that one can read off the natural laws of a world by seeing what best systematizes the entire history of it. If there are certain types of events that have no interesting pattern to them at all except for a certain limiting frequency, then the best way to systematize these will be with chancy laws. In this setting it’s not clear how one would justify the Principal Principle, or how one would claim to have knowledge about the chances.
At any rate, the Principal Principle seems to say different things on these different interpretations of chance, and it gives rise to either different justifications or different accounts of what should count as “inadmissible evidence”.