Isn't it provable that for expected utility decision making, uncertainty
on the prior is irrelevant? Can someone point to that proof?
Geoff
--- Geoffrey Rutledge, MD, PhD Director of Clinical Informatics Healtheon Corporation ---------------------------------David Poole wrote: > > Rolf Haenni wrote: > > But knowing that P(Q)=0.5 is > > clearly not the same as to know nothing about Q. > > This isn't so clear. Here is some reasoning I have used to convince > myself that these are indeed the same: > > Suppose we have a button, and when we press the button, there is one of > two outcomes, either a H or a T (just to make it look familiar). Let the > event Q be the outcome of pressing the button. > > Before we press any buttons, I ask you what will be the first outcome. > You, of course, say "I don't know". > > Suppose we press the button 4 times and observe a H,T,T,H. And then I > ask you what will be the outcome of the 5th press. You then would also > say "I don't know". > > Now suppose that we were to press the button 1,000,000 times and we > observe that half of them resulted in H and half resulted in T (and we > could not detect a pattern in the sequence). If I then ask you to say > what the outcome of the 1,000,001st press will be, you would also say "I > don't know". > > The first "I don't know" was the result of knowing nothing about Q. > The last "I don't know" was the result of knowing the P(Q) is 0.5. > > But it seems to me that these "I don't knows" are not of different > types: you have no idea whether the outcome will be H or the outcome > will be T initially or for the 1,000,001st press. [Of course you have > very different knowledge about the probability of Q, but no one would > claim any differently.] Parsimony would suggest that we don't need to > distinguish these "I don't know"s (i.e,., the prediction from ignorance > and the prediction from knowing the probability is 0.5). > > For those people who would like to distinguish ignorance for the outcome > of a binary variables and probability 0.5, I would like to know how many > different meanings are there to "I don't know" (for a binary random > variables)? (If there are a finite number (such as two) different > meanings, when in the above sequence of making predictions and pressing > the button do the meanings switch? Is the "I don't know" prediction for > the 5th toss have the same meaning as the first "I don't know" or the > same meaning as the "I don't know" for the prediction of the 1,000,001st > press? (Or isn't 1,000,000 presses enough data to warrant drawing a > conclusion about the probability?). > > I would really like to know. There has been lots of research based on > the distinction between ignorance and probability, yet this is some > reasoning I have used to convince myself that the Bayesians are right. > > Thanks, > David > > -- > David Poole, Office: +1 (604) 822-6254 > Department of Computer Science, poole@cs.ubc.ca > University of British Columbia, http://www.cs.ubc.ca/spider/poole