“So, though the asymmetry is doing some work here (the further we move above 0, the more likely that +1 rather than −1 is doing some of the work), it could still be that 23,000 is the smallest of the values I sampled”—That’s very interesting.
So I looked at the definition on Wikipedia and it says: “An estimator is said to be unbiased if its bias is equal to zero for all values of parameter θ.”
This greatly clarifies the situation for me as I had thought that the bias was a global aggregate, rather than a value calculated for each value of the parameter being optimised (say basketball ability). Bayesian estimates are only unbiased in the former, weaker sense. For normal distributions, the Bayesian estimate is happy to underestimate the extremeness of values in order to narrow the probability distribution of predictions for less extreme values. In other words, it is accepting a level of bias in order to narrow the range.
“So, though the asymmetry is doing some work here (the further we move above 0, the more likely that +1 rather than −1 is doing some of the work), it could still be that 23,000 is the smallest of the values I sampled”—That’s very interesting.
So I looked at the definition on Wikipedia and it says: “An estimator is said to be unbiased if its bias is equal to zero for all values of parameter θ.”
This greatly clarifies the situation for me as I had thought that the bias was a global aggregate, rather than a value calculated for each value of the parameter being optimised (say basketball ability). Bayesian estimates are only unbiased in the former, weaker sense. For normal distributions, the Bayesian estimate is happy to underestimate the extremeness of values in order to narrow the probability distribution of predictions for less extreme values. In other words, it is accepting a level of bias in order to narrow the range.