Probability space consists of three things: sample space, event space and probability function.
Sample space defines a set of possible outcomes of probability experiment, representing the knowledge state of the person participating in it. In this case its:
{Odd, Even}
For event space we can just take a superset of the sample space. And as our measure function we just need to assign probabilities to the elementary events:
P(Odd) = P(Even) = 1⁄2
Do I understand correctly that the apparent problem is in defining the probability experiment in such a way so that we could talk about Odd and Even as outcomes of it?
It’s an interesting question, but its a different, more complex problem than simply not knowing googolth digit of pi and trying to estimate whether it’s even or odd.
The reason why logical uncertainty was brought up in the first place is decision theory, to make crisp formal expression for intuitive “I cooperate with you conditional on you cooperating with me”, where “you cooperating with me” is result of analysis of probability distribution over possible algorithms which control actions of your opponent and you can’t actually run these algorithms due to computational constraints, and you want to do all this reasoning in non-arbitrary ways.
We don’t??? Probability space literally defines set of considered worlds.
Probability space consists of three things: sample space, event space and probability function.
Sample space defines a set of possible outcomes of probability experiment, representing the knowledge state of the person participating in it. In this case its:
{Odd, Even}
For event space we can just take a superset of the sample space. And as our measure function we just need to assign probabilities to the elementary events:
P(Odd) = P(Even) = 1⁄2
Do I understand correctly that the apparent problem is in defining the probability experiment in such a way so that we could talk about Odd and Even as outcomes of it?
The problem is “how to define P(P=NP|trillionth digit of pi is odd)”.
Interesting. Is there an obvious way to do that for toy examples like P(1 = 2 | 7 = 11), or something like that
It’s an interesting question, but its a different, more complex problem than simply not knowing googolth digit of pi and trying to estimate whether it’s even or odd.
The reason why logical uncertainty was brought up in the first place is decision theory, to make crisp formal expression for intuitive “I cooperate with you conditional on you cooperating with me”, where “you cooperating with me” is result of analysis of probability distribution over possible algorithms which control actions of your opponent and you can’t actually run these algorithms due to computational constraints, and you want to do all this reasoning in non-arbitrary ways.