## More on probability aggregation and De Finetti

A few months ago I posted a puzzle about aggregating probability estimates from different sources, and in particular how to aggregate opinions about the independence of two events.

I think I now understand the story slightly better.  I am essentially going to agree with what Terry T. said in the comments to the first post (this is my surprised face) but at the same time try to dissolve my initial resistance to talking about second-order probabilities (statements of the form “the probability that the probability is p is q….”)

To save you a click, the question amounts to:  if half of your advisors tell you that X and Y are independent coins with probability .9 of landing heads, and the other half of your advisors agree the coins are independent but say that the probability of heads is .1 for each, what should your degree of belief in X, Y, and X&Y be?  And should you believe that X and Y are independent events, a fact about which your advisors are unanimous?

The answer depends, at least in part, on what you mean by “probability” and “independence.”