A question about Lindley’s supra Bayesian method for expert probability assessment – Beragampengetahuan

Andy Solow writes:

I wonder if you can help me with a question that has been bugging me for a while? I have been thinking about Lindley’s supra Bayesian method for expert probability assessment. Briefly, the model is that, conditional on the event of interest A, the log odds ratio for an expert is normal with mean mu > 0 and variance v and, symmetrically, conditional on not-A, it’s normal with mean -mu and the same variance. When the prior probability of A is 1/2, the posterior log odds ratio given the expert’s log odds ratio q is:

(2*mu/v) * q

Lindley took v = 2*mu so that the expert’s log odds ratio is simply adopted. Now, I would have thought that mu can be viewed as a measure of expertise: the more expert the expert, the greater mu. If that’s the case, then I also would have thought that the distribution of a more-expert expert should stochastically dominate that of a less-expert expert. But this is not true under Lindley’s assumption. Stochastic dominance requires that v is a non-increasing function of mu – the simplest case being a constant v. But for mu > v/2 the posterior log odds ratio is greater than q, which doesn’t seem right either. I wonder if I am thinking about this incorrectly? Any thoughts you might have would be greatly appreciated.

My reply: I don’t know, I’ve never thought about this one! The whole thing looks kind of arbitrary to me, and I’ve never been a fan of models of “expert opinion” that don’t connect to the data used by the expert or to whatever the expert is actually predicting. But Lindley was a smart guy, so I’m guessing that the idea is more general than looks to me at first.

Can someone explain in comments?