In two recent comments [1][2], it has been suggested that to combine ostensibly Bayesian probability assessments, it is appropriate to take the mean on the log-odds scale. But Bayes’ Theorem already tells us how we should combine information. Given two probability assessments, we treat one as the prior, sort out the redundant information in the second, and update based on the likelihood of the non-redundant information. This is practically infeasible, so we have to do something else, but whatever else it is we choose to do, we need to justify it as an approximation to the infeasible but correct procedure. So, what is the justification for taking the mean on the log-odds scale? Is there a better but still feasible procedure?
An independent piece of evidence moves the log-odds a constant additive amount regardless of the prior. Averaging log-odds amounts to moving 2⁄3 of that distance if 2⁄3 of the people have the particular piece of evidence. It may behave badly if the evidence is not independent, but if all you have are posteriors, I think it’s the best you can do.
In two recent comments [1][2], it has been suggested that to combine ostensibly Bayesian probability assessments, it is appropriate to take the mean on the log-odds scale. But Bayes’ Theorem already tells us how we should combine information. Given two probability assessments, we treat one as the prior, sort out the redundant information in the second, and update based on the likelihood of the non-redundant information. This is practically infeasible, so we have to do something else, but whatever else it is we choose to do, we need to justify it as an approximation to the infeasible but correct procedure. So, what is the justification for taking the mean on the log-odds scale? Is there a better but still feasible procedure?
An independent piece of evidence moves the log-odds a constant additive amount regardless of the prior. Averaging log-odds amounts to moving 2⁄3 of that distance if 2⁄3 of the people have the particular piece of evidence. It may behave badly if the evidence is not independent, but if all you have are posteriors, I think it’s the best you can do.