Re: The danger of classical hypothesis and significance tests [was Re: MADLY AMUSED]

JoJo wrote:
S. F. Thomas wrote:
illywhacker wrote:
On May 30, 8:34 pm, "S. F. Thomas" <thomas7...@xxxxxxxxxxxxx> wrote:
illywhacker wrote:
On May 29, 7:26 pm, "S. F. Thomas" <thomas7...@xxxxxxxxxxxxx> wrote:

(( cuts ))

This contention is admittedly a bold assertion, but I may justify it very
simply by making the observation, without fear of contradiction, that
*all* of what the data say about the parameter under the maintained
model is summed up in the likelihood function.

Yes. But what about what prior knowledge has to say about the
parameter? Suppose we know it must be positive, e.g. it is a density.
Using the likelihood alone will not guarantee this fact necessarily
(many examples exist). Either this constraint must be imposed by hand
in an ad hoc way, or it must be taken into consideration in a
principled way, i.e. by using a prior. Hence the following:

I have seen this attempt before to turn the Bayesian bug into a feature. I remain unmoved. Any constraint on the parameter is in my opinion better and more honestly placed where it belongs -- within the probability model. Certainly there is no in-principle necessity to resort to a prior to achieve the same effect.

S.F.

Could you briefly explain what you consider a "bayesian bug" ?

The metaphor "bug" is used in the same way that a software programmer might use it. Turning a bug into a feature is rather like telling a customer at a restaurant that the cockroach in the soup has been added for flavor, color, and crunchiness. The Bayesian bug is the very same that bothered Rev. Thomas Bayes from the beginning (18th century), namely the lack of justification for treating the parameter constant as a random variable in its own right, and rewriting the original model as conditional probability. This idea was rejected by the now-called classical statisticians (Pearson, Neyman et al) of the late 19-th and early 20-th centuries. The neo-Bayesians of the 20-th century -- Savage, de Finetti, Jaynes et al -- tiring of the roundabout and indirect classical methods, which are prone to error, as we have seen, set about trying to justify Bayes. All manner of axiomatizations and arguments have been tried, hence such notions as personal probability, belief probability, subjective probability, belief probability, etc., all intended to justify mixing the frequency probability of the original model, with some notion of probability inhering in our uncertainty regarding the unknown constant parameter sought to be estimated. In the early years, the problem was rightly considered a problem -- why else expend the enormous intellectual effort that has been expended. More recently, some have taken to asserting the Prior, and the probabilistic treatment of unknown constants, to be a feature, as we just saw in this thread. In fact, the Prior is *in principle* irrelevant to the problem of inference because the whole inferential import about what the data say about the true but unknown parameter value is summed up entirely in the Likelihood. Therefore, I maintain, the notion of prior and etc. remain essentially a bug in the inferential soup served by the Bayesians. Notwithstanding the colorful metaphor, I say that with the utmost respect -- but my chemistry teacher taught me a long time ago to sit always humbly before fact. The insight I hope to have brought to the debate is the idea that likelihood -- also fuzziness, also possibility -- is a categorically different form of uncertainty in its own right, and one may manipulate the likelihood function directly, to achieve inferential results sought, without taking a detour through the land of Prior. I know the Bayesians think they have constructed an impregnable fortress with their notions of rationality, consistency and etc., but as long as their axiomatization does not allow for or explicate the notion of a "fuzzy probability", I remain unmoved by the rationality claims. (Here is where the issue of real numbers and the total ordering axiom come into play, but that is another story which I will refrain from entering here.)


I was following this thread but am a little buffled, not sure what are your reservations about bayesian approach.

I hope the above quick recitation has helped.

Regards,
S. F. Thomas
.