
Re: Trying to understand Bayes and Hypothesis
Posted:
Feb 20, 2013 1:31 PM


On 20130219, 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, prebuilt 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/NeymanPearson issue. I am a deep believer in doing what works. I also have a strong preference for things like the ttest (properly used) as it requires no thinking and no work. Costbenefit should never be ignored.
Costbenefit should not be ignored, and in fact the basis of decision theory is that of comparing risks. Simple axioms of selfconsistent 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)4946054 FAX: (765)4940558

