This is a followup on the preceding post.
1. Whenever the rational credence of p is 0.5 on some evidence base E, at least 50% of human agents who assign a credence to p on E will assign a credence between 0.25 and 0.75.
2. The log-odds of the credence assigned by human agents given an evidence base can be appropriately modeled by the log-odds of the rational credence on that evidence base plus a normally distributed error whose standard deviation is small enough to guarantee the truth of 1.
3. Therefore, if I have no evidence about a proposition p other than that some agent assigned credence r on her evidence base, I should assign a credence at least as far from 0.5 as F(r), where:
- F(0.5) = 0.5
- F(0.6) = 0.57
- F(0.7) = 0.64
- F(0.8) = 0.72
- F(0.9) = 0.82
- F(0.95) = 0.89
- F(0.98) = 0.95
- F(0.99) = 0.97
4. This is a pretty trusting attitude.
5. So, it is rational to be pretty trusting.
The trick behind the argument is to note that (1) and (2) guarantee that the standard deviation of the normally distributed error on the log-odds is less than 1.63, and then we just do some numerical integration (with Derive) to compute the expected value of the rational credence.