Just out of curiosity, under what circumstances would the sentence "I am not
thinking about the tooth fairy" be true? You could concoct a great bogus
"proof" on the grounds that anyone who denies thinking about the tooth fairy
must be thinking of the tooth fairy, so therefore everyone must always be
thinking of the tooth fairy, which would constitute pretty strong evidence
(given reasonable priors) that the tooth fairy must exist. Of course, this
applies equally to God and to "blue Latin-speaking turtles" ...
:-)
At 7/6/99 04:01 PM, Bob Welch wrote:
>>The problem is that people are often very reluctant to admit the
>>modeling assumptions that they have made.� In the tooth fairy example,
>>one might be tempted to claim that the original model assumed only that
>>the existence of tooth fairy would predispose people to wonder about the
>>tooth fairy (more so than they would otherwise wonder) and then argue,
>>upon the arrival of data showing wondering, that the tooth fairy is more
>>likely to exist.� What's wrong with this?� The model does not mention
>>other causes of wondering and, thus, implicitly encodes that *any* cause
>>of wondering must be the tooth fairy.
>>
>>We have a situation where the tooth fairy believer is claiming an
>>inference from a property of the a hypothetical tooth fairy + data to
>>the an increased posterior on the tooth fairy.� However, when we add in
>>the implicit modeling assumptions, we see that it is really an inference
>>from the assumption that the tooth fairy is the only cause of a
>>phenomenon + data to the existence of the tooth fairy.� While the first
>>version purports to be making scientific evaluation about the existence
>>of the tooth fairy, the second demonstrates that the modeler has simply
>>codified his prejudices in the model and is making a rather
>>uninteresting claim that is primarily about his own prejudices.
>
>But within the context of Bayes rule, the modeler must have made an
>assessment of P(B | A) where A = "tooth fairy exists" and B = "I am thinking
>about the tooth fairy."� This is a 2x2 matrix of values. That assessment
>requires the modeler to consider not only the probability that he would be
>thinking about the tooth fairy when the tooth fairy exists, P(B=T|A=T),� but
>to also consider the probability of not thinking about the tooth fairy when
>the tooth fairy exists,� P(B=F|A=T),� or why p(B=T|A= T) is less than one.
>In addition the assessment requires thinking about the tooth fairy when the
>tooth fairy doesn't exists P(B=T!A=F) . And that requires the modeler to
>consider all the other possible causes.� He doesn't have to make the other
>causes an explicit part of his model.� They are in there implicitly� and the
>use of Bayes rule requires that they be part of the calculation.� The values
>of the terms in the likelihood matrix p(B | A) are not implicit assumptions.
>The modeler is required to make an assessment.� This is not a closed world
>modeling environment.
>
>Now it may be that in order to make that assessment, he needs to make the
>other factors an explicit part of the model, expanding the original Bayesian
>network.� That is his choice based on his comfort with the assessments he
>has made. But in any case he must give values for all the terms in the cp
>table of B.
