The principle of indifference states that:
"in the absence of any relevant evidence, agents should distribute their credence (or 'degrees of belief') equally among all the possible outcomes under consideration."
This is for determining epistemic probability. I.e., these are not repeatable events and the possibilities are not drawn out of a bag. Rather, in this scenario there is and always was a probability of 1 for the actual outcome and a probability of 0 of all others, it is our ignorance which forces us to ascribe a lower probability to a given outcome.
Suppose I have a set of N elements. The nth element is defined as a two element sub-set of the form (n, B) for all n<N, where n is simply its index and B is a Boolean, e.g. the 3rd could be (3, 0) or (3, 1). Lets suppose one of these elements is "true", and there is no available evidence to determine which is true. By the principle of indifference, each possibly outcome is equally likely. We can also say that the probability of B=1 is 50% by the same principle.
Now let's say that B=1 only when n is a multiple of 3. If we apply the principle of indifference to each possible element and summing those probabilities, we find that the likelihood of B=1 is around 1/3 if N is large. This implicitly assumes that the discrepancies between the number of elements with 1 and 0 serves as evidence for B. But this could just as easily flipped the other way around, treating B with indifference and using that as evidence that the likelihood of n being a given multiple of 3 is lower than it being a given number which is not a multiple of 3. We arrive at different probabilities for the same question.
If you are not convinced that we can flip the situation around, then this example may make the problem more obvious. I have either a cube or a sphere, and each face can take one of 3 colours. Clearly there are more possible cubes than spheres, so should I expect to have a cube rather than a sphere or is the probability of having either equally likely?
This is just a less abstract version of the previous problem (with the variable B signifying a cube or a sphere), albeit in different proportions. It seems people are more willing to treat the variable B with indifference in this more concrete example (i.e., 1/2 chance B=1), whereas with the numerical example they are more willing to treat the individual elements with indifference (i.e., ~1/3 chance B=1).
I have three questions with regards to this apparent paradox:
- Which "level" should we treat with indifference and why?
- If it is situation dependent, then how does context determine which we choose?
- Why does our intuition seem to change between the two mathematically equivalent examples?