The interpretation of the results
of any analysis cannot be done in a vacuum, rather it strongly depends of the
tools used to reach it. The choice of tools should be justified by their
suitability to the task, and their logical treatment of the data. Any
person dealing with data and trying to extract meaning from it normally is very
careful of the 'validity' of the data he/she uses, and that is good.
Unfortunately, few people are so careful at the time to choose the
'tools' to interpret that same data. No body flinches when an 'expert'
announces that the most probable mass of such and such particle is minus
something, like saying that the most probable state of a person is 'unborn' or
such other nonsense.
For my part, I always had problems
when physical meaning is attached to concepts like infinity. Conversing
with experts, many times the made me feel like an obtuse simpleton until I came
across the following statement from Gauss:
"I protest against the use of infinite magnitude as something
accomplished, which is never permissible in mathematics. Infinity is merely a
figure of speech, the true meaning being a limit."
I have a similar problem when
trying to follow the reasoning behind 'ortodox' probability theory with all the
paradoxes that it appear to engender. Interestingly enough, the
prevailing idea that probability is a theory of chance denude of logic is a
late comer. The pillars of probability theory (Gauss, Laplace, etc.) look
at it more as scientific inference that complies with the rules of logic rather
than the result of flipping coins. Perhaps people continue to use the
orthodox frequentist approach because the great amount of
ad hoc devices constructed during the
years to deal with specific situations. And this is done some times
applying tools in a particular case when the tools were designed for some other
field without asking the question of the validity of such usage. No
wonder that some obtained results are inconsistent or absurd. It is
easier to apply a 'formula' without much thought than to think carefully about
the problem at hand. So, people continue using cabalistic incantations to
solve their problem even when they know that the tool the use is suspect and
totally useless when applied to other category of problems or even to a more
complex problem belonging to the same
category.
Fortunately, there is a way out: A Probability
Theory based in sound, logical rules of inference. And that happens to
provide two powerful tools: The Bayes' Theorem and the Principle of Maximum
Entropy. These and the consistent application of logic and rules of
inference is all that is needed to solve the most complex problem without fear
of arriving to inconsistent or absurd results. |