Let’s say Alice and Bob both estimate a 80% chance of rain today. However, Alice has already checked a high quality weather report, and Bob hasn’t and is going on vibes. When the time comes to leave the house with or without an umbrella Bob will probably choose to check the weather, and update his expectations accordingly, because it’s easy, and makes it more likely that he will carry an umbrella if and only if it will rain. Whereas Alice will have to make a decision based on her current beliefs—getting even better information will be hard and expensive.
Bob took advantage of the fact that his probability distribution was more spread out than Alice’s. His 80% included both the possibility that the weather report would rightly say it is going to rain and the possibility that it would wrongly forecast a sunny day. He can look at the forecast and update, and do better as a result. Alice’s 80% was already conditioned on the dreary forecast, and thus narrower, and less susceptible to change.
To summarize, the information as to whether you are confident or unsure of your belief is in the shape of your probability distribution (but not in the number 80%) and if you are unsure you often have a cheap and valuable option to learn more.
(What Bob definitely wouldn’t do is bet Carol the weather reporter that it will rain today at 4:1 odds. Despite betting odds supposedly reflecting your beliefs, don’t bet against people you know have better information!)
Let’s say Alice and Bob both estimate a 80% chance of rain today. However, Alice has already checked a high quality weather report, and Bob hasn’t and is going on vibes. When the time comes to leave the house with or without an umbrella Bob will probably choose to check the weather, and update his expectations accordingly, because it’s easy, and makes it more likely that he will carry an umbrella if and only if it will rain. Whereas Alice will have to make a decision based on her current beliefs—getting even better information will be hard and expensive.
Bob took advantage of the fact that his probability distribution was more spread out than Alice’s. His 80% included both the possibility that the weather report would rightly say it is going to rain and the possibility that it would wrongly forecast a sunny day. He can look at the forecast and update, and do better as a result. Alice’s 80% was already conditioned on the dreary forecast, and thus narrower, and less susceptible to change.
To summarize, the information as to whether you are confident or unsure of your belief is in the shape of your probability distribution (but not in the number 80%) and if you are unsure you often have a cheap and valuable option to learn more.
(What Bob definitely wouldn’t do is bet Carol the weather reporter that it will rain today at 4:1 odds. Despite betting odds supposedly reflecting your beliefs, don’t bet against people you know have better information!)