thanks to Ursula, Jonathan, and Wang Pei for their help to clarify the
problem of representing total ignorance.
Let me add the following answer to David Poole's question:
DAVID POOLE wrote:
>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)?
There are two different meanings of saying "I don't know" to the outcome of
a (binary) variable Q:
1) I DON'T KNOW the outcome of Q, but I KNOW (from my experience on
earlier or similar events or processes) a prior probability P(Q).
2) I DON'T KNOW the outcome of Q, and I have no experience allowing
me to assign any prior probability.
Total ignorance corresponds to the 2nd case.
Let me illustrate this by considering the question "Does God exists?".
There are two possibilities G and ¬G (G = "God exists", ¬G = "God does not
exist"). Following the arguments of David Poole, it would be necessary to
assign a prior probability, that is to say
P(G) = P(¬G) = 0.5,
which is completely absurd (even to speak about "probability that God
exist" is absurd). In contrast, I think it makes perfectly sense to say
that
Bel(G) = 0, Pl(G) = 1,
Bel(¬G) = 0, Pl(¬G) = 1, and
Bel(G or ¬G) = Pl(G or ¬G) = 1,
which corresponds the agnostical point of view of a person who thinks that
nothing can be known about God (= total ignorance).
A similiar situation is the example given by Jonathan Weiss: at the
beginning, when nothing is know about the deck of cards and about the color
of the first card dealt, I think it does make sense to assign probabilities
P(red), P(green), etc... In contrast, it makes perfectly sense to say
Bel(red) = 0, Pl(red) = 1
Bel(green) = 0, Pl(green) = 1
...etc,
Bel(red OR green OR blue OR yellow OR "something else") = 1.
Note that from the point of view of probabilistic argumentation systems,
Bel(Q)=0 means that there are no arguments in favor of Q, while Pl(Q)=1
means that there are no arguments against Q. In my eyes, this reflect
perfectly to nature of total ignorance.
Finally, let me add what Lao Tse said: "Knowing Ignorance is Strength".
Best wishes,
Rolf Haenni
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* Dr. Rolf Haenni __/ __/ __/ __/ _______/ *
* Institute of Informatics (IIUF) __/ __/ __/ __/ __/ *
* University of Fribourg, Switzerland __/ __/ __/ __/ _____/ *
* Phone: ++41 26 300 83 31 __/ __/ __/ __/ __/ *
* Email: rolf.haenni@unifr.ch __/ __/ ______/ __/ *
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