The question was whether you could say that you believed in something that you thought occurred with less than 0.5 probability. If you thought you would crash your car with less than 0.5 probability, you may still wear a seatbelt, but you wouldn’t say that you believed you would crash your car.
Above you wrote:
We now have a situation where p(blue) < .5 but where you nevertheless “believe that” the diamond is in the blue box, insofar as that is the box you’d pick.
In your three-box example, if you believed the diamond was in the blue box with p=0.4, and the other two with p=0.3 each, it would sound very strange to me to say “I believe the diamond is in the blue box.” Instead, I would say “I think it’s most likely that the diamond is in the blue box.”
The use of the word “believe” doesn’t correspond to a single probability level, and as such, isn’t very Bayesian. For instance, say there is a lottery with one million tickets, and you have one ticket. Do you believe you will not win? No, and that seems true no matter how many tickets the lottery has.
Essentially, use of the word “believe” indicates that you’re talking about an axiom, a statement that you’re using as a further assumption without questioning or looking at the probabilities.
The question was whether you could say that you believed in something that you thought occurred with less than 0.5 probability. If you thought you would crash your car with less than 0.5 probability, you may still wear a seatbelt, but you wouldn’t say that you believed you would crash your car.
Above you wrote:
In your three-box example, if you believed the diamond was in the blue box with p=0.4, and the other two with p=0.3 each, it would sound very strange to me to say “I believe the diamond is in the blue box.” Instead, I would say “I think it’s most likely that the diamond is in the blue box.”
The use of the word “believe” doesn’t correspond to a single probability level, and as such, isn’t very Bayesian. For instance, say there is a lottery with one million tickets, and you have one ticket. Do you believe you will not win? No, and that seems true no matter how many tickets the lottery has.
Essentially, use of the word “believe” indicates that you’re talking about an axiom, a statement that you’re using as a further assumption without questioning or looking at the probabilities.