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Re: Trying to understand Bayes and Hypothesis
Posted:
Feb 20, 2013 1:31 PM
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On 2013-02-19, Dave <divergent.tseries@gmail.com> wrote:
> I sadly work with the Cauchy distribution all the time. It is the natural distribution function for the data I work with, although it is part of a mixture distribution.
> People regularly mistake Cauchy driven data for normally driven data
as people do not realize how common it is nature. Poisson's comment
to Laplace about only needing to footnote the existence of the Cauchy
distribution as it will not be encountered in practice hasn't helped any.
> One thing that Bayesian statistics lacks, that Frequentists have in abundance is a nice, pre-built set of diagnostic tools.
The alchemists and other medieval "natural philosophers" likewise
had such tools. As with the case then, these tools are often very
wrong. The situation is not as simple.
One which really needs to be eliminated is "statistical sibnificance".
There ARE cases where it makes some sense, but the point null hypothesis
as stated is always false, and the real question is what action should
be taken. A statistically significant difference may be of no practical
or even theoretical importance, and a statistically insignificant difference
may be very important. This simple tool is like a loaded gun in the hands
of an ignoramus as to how guns work.
> I am philosophically neutral on the Bayesian/Fisherian/Neyman-Pearson
issue. I am a deep believer in doing what works. I also have a strong
preference for things like the t-test (properly used) as it requires no
thinking and no work. Cost-benefit should never be ignored.
Cost-benefit should not be ignored, and in fact the basis of decision
theory is that of comparing risks. Simple axioms of self-consistent
behavior, see for example my paper is _Statistics and Decisions_ 1987,
show that one must compare integrated risks. IF one has a good idea
of the integrating factor, the optimal solution is Bayesian, again
IF it can be computed. Those are big ifs, and approximations can
be made.
> I do think that part of the genius of Frequentist statistics is its
decision tree or algorithmic nature. If you encounter data like this,
then analyze it like that. If some different assumption is present
or a warning statistic clicks on as a problem then do some other thing
instead as a robustness check.
There is nothing here which is foreign to decision theory, definitely
including Bayesian.
> But a Frequentist is presuming to know the true model of the world
and so has to be able to do this. A Bayesian is inducing the model from
the data. Still, it wouldn't kill us to have a decision tree like...if
your problem looks like this then the following likelihood functions
could work to model the problem. And further, the following posterior
simulations do X,Y,Z in practice. Except for the conjugate families,
we don't really have this.
Your statements are incorrerct. Classical philosophy of science
has the scientist deriving the theory from data. This is false;
one can only choose between theories from the data. The ideal
decision make has all models in his mind and chooses between them;
we are not ideal, so we have to put some weight on "something else".
Again, this means that we imperfect decision makers have to use
prior Bayes methods which allow for such errors.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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