let P(R|Q)=0.9 and P(R|¬Q)=0.8. In such a case, knowing that either Q or ¬Q
is true, belief functions already allow to deduce Bel(R)=0.72 and
Pl(R)=0.98, which makes perfectly sense in my eyes.
This didn't make sense in Ross Schachter's eyes (nor in mine), so Haenni
justified this with an argument that introduced additional propositions A1 and
A2 that were not part of the original problem. As far as deriving bounds goes,
I believe his reasoning is correct, but his bounds are still too loose. He
correctly uses P(A1 and A2) as a lower bound for P(R). However, we don't need
(A1 and A2) to be true for R to be true -- it suffices that
(A1 and Q) or (A2 and ~Q)
be true, a condition that is weaker than (A1 and A2). Call the above
proposition B. We have
(A1 and A2) implies B
and so
P(A1 and A2) <= P(B).
Now it is easily proven that P(R) must lie between 0.8 and 0.9. Since it's not
clear whether Q is intended to be a repeatable event, and Haenni's remarks lead
me to believe that he is a firm frequentist, I have to be careful about bringing
P(Q) into the discussion with any value other than 0 or 1. So let's look at two
cases:
1) Q is a non-repeatable event.
Then the situation is very simple. If Q is in fact true, then P(Q) = 1 and
hence P(R) = P(R) / P(Q) = P(R|Q) = 0.8. If Q is in fact false, then P(~Q) = 1
and hence P(R) = P(R) / P(~Q) = P(R|~Q) = 0.9. Thus
P(R) = 0.8 or P(R) = 0.9
and hence
0.8 <= P(R) <= 0.9.
So Haenni's theory appears to be losing some information, by giving
unnecessarily loose bounds on P(R). It also loses information in that it can't
even represent what we know about P(R) -- that it is *exactly* 0.8 or *exactly*
0.9, and not any value in between these two.
2) Q is a repeatable event.
Then
P(R) = P(R,Q) + P(R,~Q)
= P(R|Q) * P(Q) + P(R|~Q) * P(~Q)
= 0.8 * P(Q) + 0.9 * (1 - P(Q))
and so, regardless of the value of P(Q), we know from 0 <= P(Q) <= 1 that 0.8 <=
P(R) <= 0.9. Again, Haenni's theory is losing information by giving
unnecessarily loose bounds.