Ronald,
>For example, I might have the prior
>assumption that if the tooth fairy existed, she would predispose people
>to ask questions about the tooth fairy (more than they would be
>predisposed to do otherwise). Now I entertain the hypothesis that the
>tooth fairy exists. The fact that I have asked the question is evidence
>that I am predisposed to ask the question. This increases my posterior
>on the existence of the tooth fairy.
Exactly what is wrong with this argument? It is a perfectly coherent set
of beliefs (which I do not happen to share) that might plausibly be held by
someone who places credence in the existence of the tooth fairy, and
suspects that if she existed she would want us to know and would somehow
have the power to "wire" us to want to know.
Let F be the hypothesis that the tooth fairy exists, and W be the
hypothesis that I wonder about her existence. What is wrong with my
constructing the Bayes net F-->W and assigning probabilities? That's a
perfectly fine model for me if it represents my beliefs about the tooth
fairy.
To avoid double-counting, I have to assign prior probabilities for
existence and non-existence that do not already incorporate the evidence
that I am wondering, which may be difficult psychologically. That I'm even
entertaining the hypothesis and assessing a prior means I'm already
wondering. But one could imagine asking oneself, "What would my prior
probability be in the existence of some being I have never wondered about?"
>Indeed, every time I ask the
>question my posterior increases.
Sure, why not? The harder I wonder, the stronger the evidence (to me) that
the thing I'm wondering about exists. You could build your model that way.
You could also build your model so that given one instance of wondering,
additional instances had no further evidential support. It all depends on
your model for how the tooth fairy induces wondering in the people she
wants to believe in her.
>This is clearly bogus since I can use
>this line of reasoning to justify anything.
No! I can't justify the above argument to myself not because it's a bogus
line of reasoning, but because I don't believe the F-->W link, and also
because I place a miniscule prior on F being true.
If those are your beliefs, then you have a justification for them. I
disagree strongly with the hypothetical individual whose model we are
discussing here, but she has the right to have her model without my calling
it bogus and pretending there is some mathematical reason why she's not
allowed to believe in it. If her models are all this strange, I suspect
her ideas will not percolate very far into the collective consciousness,
and some shyster might be able to figure out how to make a lot of money
from her, but let her have her ideas. (Gregory Bateson called the "New
Age" belief system the essential element of diversity that may prevent the
Western scientific worldview from ossifying into rigidity and
self-destructing. We may need tooth fairy believers to keep us honest.)
Patrick Glynn uses just this argument for the existence of God in his book
"God: The Evidence," but his Bayes net (he doesn't write it as a Bayes
net; he uses those "word arguments" you so dislike) has G instead of F as
the root node. I myself find a positive G-->W link far more plausible than
a F-->W link, although not as strong as Glynn finds it. You clearly don't
see any difference between the F and G hypotheses. Hence, we disagree on
the import of the evidence. Vive la difference. But neither of us is
RIGHT in any provable sense.
>The hypothesis, along with
>the other assumptions I have made form a vicious circle.
Not true. I've just argued from prior to posterior. Nothing circular.
Our tooth fairy believer's Bayes net is a perfectly good deductive
argument, and she may apply it to the world if she so chooses.
Kathy
