
Re: Trying to understand Bayes and Hypothesis
Posted:
Feb 22, 2013 3:09 AM


"Dave" wrote in message news:7e66c68aac394207a39903d64e0277fe@googlegroups.com...
(1) Theory says the "errors" should be normally distributed and no one argues that a variety of goodness of fit measures reject it at p<.001 or wherever the table stops.
(2) Theory says I should be able to minimize variance choosing an expectation or maximize an expectation choosing a variance. Of course you cannot do that with a Cauchy distribution.
==============================================================================================
Step (2) is incorrect, given the results of step(1). Given step (1), "theory" says either:
(a) Chose an appropriate likelihood function, based on an acceptable distribution. Use a large sample argument to justify a chisquared test based on a likelihood ratio test.
(b) Choose an appropriate objective function (goodnessoffit measure), such as a mean absolute difference. (Although this might need to be modified if you are fitting both location and scale.) Construct a test statistic based on this objective function, such as the improvement in the objective function on moving to the wider model. Construct critical values for the test statistic by undertaking a simulation study based on what you think are acceptable null distributions.
If you were happy enough to do a Bayesian analysis, you might note that several recent works have been implemented with structures where the "normal distribution" assumption has been replaced by a Student's t distribution with fixed but low degrees of freedom, which includes the Cauchy distribution. Hence there is a good chance that you could find a Bayesian analysis package that includes this facility, and this might prove a viable route for you. Of course, you might find a"frequentist" package to do something similar, if you need to look for preexisting code ... you might look under "general linear model".
David Jones

