Hal, the surprising part of the beans-in-a-jar problem is that the guessers must collectively act as an unbiased estimator—their errors must nearly all cancel out, so that variance accounts for nearly all of the error, and systematic bias for none of it. Jensen’s Inequality does not account for this surprising fact, it only takes advantage of it.
Robin, I don’t claim to frame any general rule for compromising, except for the immodest first-order solution that I actually use: treat other people’s verbal behavior as Bayesian evidence whose meaning is determined by your causal model of how their minds work—even if this means disagreeing with the majority. In the situation I framed, I’d listen to the other math students talking, offer my own suggestions, and see if we could find the hidden gotcha. If there is a principle higher than this, I have not seen it.
Hal, the surprising part of the beans-in-a-jar problem is that the guessers must collectively act as an unbiased estimator—their errors must nearly all cancel out, so that variance accounts for nearly all of the error, and systematic bias for none of it. Jensen’s Inequality does not account for this surprising fact, it only takes advantage of it.
Robin, I don’t claim to frame any general rule for compromising, except for the immodest first-order solution that I actually use: treat other people’s verbal behavior as Bayesian evidence whose meaning is determined by your causal model of how their minds work—even if this means disagreeing with the majority. In the situation I framed, I’d listen to the other math students talking, offer my own suggestions, and see if we could find the hidden gotcha. If there is a principle higher than this, I have not seen it.