What if we restrict GM's perspective or authority strictly to what the characters are experiencing? Instead of having him deal with the whole setting and situation, he'll only portray and describe whatever gets to affect or be perceived by the characters. Or he might even "portray" the characters' senses instead of the world. That means that even facts perceived by the characters might be false, but the GM doesn't know this until the falseness or trueness of a fact directly affects the characters.
This might mean that, when the characters hear noises in the night and think there's a predator after them, not even the GM knows for sure if that's true or not. He will pick the odds of a predator actually chasing them, but won't actually roll until the hypothetical creature is in the best spot for an attack. Until then, he should be vague in his descriptions, to hint at both things being possible (to a different degree, perhaps).
The GM might switch roles, portraying the characters' perception during sessions, and administrating the whole setting and situation between sessions.
I think this could even be done in a transparent way, by having players know the odds of possibilities and affect them with mechanics (make a possibility 10% more likely with currency expenditure or powers, for example), or even have the players set the odds in certain situations: an excellent investigator might choose the odds of someone being guilty of a crime, for example (as long as he isn't able to just pick 100%).
I'm interested in the effects this might produce in how the GM describes the situation, as well as how would players behave in such a game. Would this produce an effect of uncertainty, even in the GM? Is it better or worse that players know and interact with these odds, to achieve a sense of uncertainty that doesn't feel arbitrary? Also, how would GM prep look in such a game?