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