Say I have two urns:
One is filled with numbered green balls, all of which lie in the range of 1 to 100.
The other is filled with numbered red balls all of which lie in the range of 1 to 500.
I draw a sequence of balls from a single urn, announce the numbers and I then ask you what color you think the urn is and to what degree of probability. Notice though, that the distribution of numbers is not specified as uniform within that range, it’s possible that there are twice as many odd numbers as even numbers for example.
Obviously, if there is a single ball >= 101, then you can assert with 100% probability that the urn is green. However, there’s no possible sequence of balls that could definitively prove a red urn.
It seems, without knowledge of the distribution, it’s impossible to be able to give precise probabilities for any other outcome. However, I could argue for certain properties of the probability. For example, it seems intuitive that the drawing of an additional ball under 100 can never decrease my posterior probability of a red urn, it can only leave it unchanged or increase it.
Also, it seems intuitive that drawing of “1, 3, 3, 2, 4, 3, 6, 8” provides less evidence of a red urn than “99, 93, 100, 96, 99, 98, 96, 100” because the first sequence can be easily explainable under both the green urn and red urn scenario but the second distribution can only be easily explainable under the red urn scenario. In fact, you might even be able to generalize it and say “the closer a ball is to 100, the greater it increases the posterior probability”
I view this as analogous to the problem of the existence of god. Under a certain formulation of god*, the space of possible universes in which god does not exist is a strict subset of the space of possible universes in which god does exist. It’s therefore strictly impossible to prove that god does not exist. However, you can view each observation as drawing a ball out of an urn and each observation can be consistent with an atheistic or supernatural interpretation of the world.
I think this analogy has helped me clarify both how to correctly think and more importantly, how to explain to others how to think about the existence of god. Yes, it’s strictly true that you can’t disprove the existence of god but by putting it into concrete terms of the urns, they can understand how evidence can be gathered for either side.
However, the urn model is still not perfect. It seems intuitively obvious to me that watching a rock fall over and over again should only have an insignificant effect on my posterior probability. Additionally, the space of supernatural universes is not 5x bigger than atheistic universes, it’s many googleplexes bigger, perhaps possibly infinitely bigger. However, it also seems intuitive to me that the size of the range of the urn should not have an effect on how to evaluate the posterior probabilities.
- In this case, a being who possesses powers outside of the realms of scientific law but not tied to any particular religious interpretation of God.