If I had a die which rolled an average of 3, my expectation for the individual probabilities would depend on my priors for why the die might be biased. For example, a die that never landed on 6 but landed on the other numbers equally, would average to 3 and be an excellent cheater’s die. A die with a slightly increased probability to roll 1 would be a good candidate for manufacturer’s defect. Could I truly have no priors on how a die came to be biased?
Edit: I just realized that I put the wrong number of sides on my die, since you were talking about a 4 sided die.
You could if you were the robot we are constructing.
But yes, if you are a human you’ll have some complicated information about how dice work. If you want to put this knowledge aside and see things from the robot’s perspective, you might try replacing “die” with “object 253,” “the die lands on side 1” with “event 759,” and so on.
But words have meaning—you shouldn’t neglect that a “die” is a “3-dimensional object with n sides, usually designed such that each has equal probability of ending up when rolled”. My objection would vanish if instead of a die it were a “random number generator returning integers from 1 to 4 with an average of 3”
In this case, the generator has an unknown distribution biased in some way toward the higher numbers. This is a problem because it is highly unlikely that my estimate of the probabilities matches the actual probabilities, and furthermore someone else might know the actual probabilities and be able to win bets against me.
If I had a die which rolled an average of 3, my expectation for the individual probabilities would depend on my priors for why the die might be biased. For example, a die that never landed on 6 but landed on the other numbers equally, would average to 3 and be an excellent cheater’s die. A die with a slightly increased probability to roll 1 would be a good candidate for manufacturer’s defect. Could I truly have no priors on how a die came to be biased?
Edit: I just realized that I put the wrong number of sides on my die, since you were talking about a 4 sided die.
You could if you were the robot we are constructing.
But yes, if you are a human you’ll have some complicated information about how dice work. If you want to put this knowledge aside and see things from the robot’s perspective, you might try replacing “die” with “object 253,” “the die lands on side 1” with “event 759,” and so on.
But words have meaning—you shouldn’t neglect that a “die” is a “3-dimensional object with n sides, usually designed such that each has equal probability of ending up when rolled”. My objection would vanish if instead of a die it were a “random number generator returning integers from 1 to 4 with an average of 3”
In this case, the generator has an unknown distribution biased in some way toward the higher numbers. This is a problem because it is highly unlikely that my estimate of the probabilities matches the actual probabilities, and furthermore someone else might know the actual probabilities and be able to win bets against me.