Q: let’s say I offer you a choice between (a) and (b).
a. Tomorrow morning you can flip that coin in your hand, and if it comes up heads, then I’ll give you a dollar.
b. Tomorrow morning, if it is raining, then I will give you a dollar.
If you choose (b) then your probability for rain tomorrow morning must be higher than 1⁄2.
Well… kinda. It could just be that if it rains, you will need to buy a $1 umbrella, but if it doesn’t rain then you don’t need money at all. It would be nice if we had some sort of measurement of reward that didn’t depend on the situation you find yourself in. Decision theorists like to call this “utility.”
I’m not sure if it’s silly to try to define probabilities in terms of decision theory rather than vice versa. ET Jaynes defines probabilities as real numbers that we assign to propositions representing a “degree of plausibility,” and satisfying some desiderata. Eli has lately been talking about probabilities in terms of the fraction of statements assigned that probability which are true, but I don’t think he considers this a definition of probability (I hope not; it would be a bad definition).
Anyway, I’ll say that what makes something a probability is not any property of the thing it references; it’s what you do with it. If you use it to weight hypotheses in expected utility calculations which determine your actions, then it’s a probability.
Q: let’s say I offer you a choice between (a) and (b).
a. Tomorrow morning you can flip that coin in your hand, and if it comes up heads, then I’ll give you a dollar. b. Tomorrow morning, if it is raining, then I will give you a dollar.
If you choose (b) then your probability for rain tomorrow morning must be higher than 1⁄2.
Well… kinda. It could just be that if it rains, you will need to buy a $1 umbrella, but if it doesn’t rain then you don’t need money at all. It would be nice if we had some sort of measurement of reward that didn’t depend on the situation you find yourself in. Decision theorists like to call this “utility.”
I’m not sure if it’s silly to try to define probabilities in terms of decision theory rather than vice versa. ET Jaynes defines probabilities as real numbers that we assign to propositions representing a “degree of plausibility,” and satisfying some desiderata. Eli has lately been talking about probabilities in terms of the fraction of statements assigned that probability which are true, but I don’t think he considers this a definition of probability (I hope not; it would be a bad definition).
Anyway, I’ll say that what makes something a probability is not any property of the thing it references; it’s what you do with it. If you use it to weight hypotheses in expected utility calculations which determine your actions, then it’s a probability.