Ron:
1. What is wrong with circular reasoning.
In a deterministic logic model, where propositions are either true or false,
circular reasoning can lead to flip/flops in the truth of a proposition,
each time through the cycle, the proposition changes from true to false or
false to true. The model is inconsistent and consequently, anything can be
proven.
But in a probability model, this is not necessarily a problem. Suppose I do
give a positive prior to the existence of the tooth fairy, small, but
greater than zero. And in my model the conditional probability that someone
will ask me about the existence of the tooth fairy if the toot fairy exists
is also positive (but less than one). However I also give positive
probability to getting that question from someone when the tooth fairy
doesn't exist. Now in this model my posterior on the tooth fairy increases
when someone asks the question and that increases the chance that the
question will be asked. (However the likelihood's do not change). As more
and more people ask the question, the posterior on the tooth fairy's
existence increases and converges to a limit, but the limit won't be one.
Circular? sure. so what. Can I prove anything? Not really.
2. Closed worlds. A probability model is by its nature is an open world
model. By stating a proposition such that the probability that I will get a
question about the tooth fairy if it exists is less than 1, I am also
stating that there are other reasons that I might not get asked could that
question when it exists -- a false negative. And by stating that the
probability that question won't be asked when the tooth fairy doesn't exist
is less than one I am allowing for all sorts of reasons why I might get
asked the question -- a false positive -- including the possibility that
there are a lot of crazy people who believe in the tooth fairy.
So in a probability model, many other possibilities are implicitly allowed
by giving a probability value that is positive but less than one. Applying
Bayes rule is circular? Ok. Call it that if you will. But I don't see it
as a problem unless all my probability assessments are 0's or 1's.
Bob Welch
