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On 8/31/05 4:45 AM, "[EMAIL PROTECTED]" <[EMAIL PROTECTED]> a écrit : >> You are also using a non-traditional definition of 'statistical bias'. The >> Gauss-Markov theorem guarantees that if condition (1) above holds [even if >> (2) and (3) are violated], then the least-squares solution is unbiased. Here >> I am using the standard def of 'bias' in statistics: > > I'm using the same definition - more or less ;o) Only from a Bayesian > viewpoint, what corresponds to a 'point estimate' would be the expectation > value of the posterior probability for which you, of course, need the > distribution. Well, I can see how there is a Bayesian analog of 'bias' (though I have never seen a precise def), but it is still a very different concept from the frequentist usage of 'bias', which is meaningless for a Bayesian. > If you integrate over a different distribution, you'll often get a > different estimate. If you only _use_ the mean and the variance, you are > in effect committing yourself to a normal distribution. > > Ok. We have to separate two issues. One is the set of assumptions laid out > by a traditional LS derivation and the other is the 'biased view' that > Bayesian Theory is the only reasonable scientific inference machine. You > are right that LS does not explicitly make any assumptions about the error > distribution, however if you analyse standard ad hoc traditional > techniques from the Information Theory and Bayesian point of view (which > can be done with some care) you often reveal the real hidden underlying > assumptions on which you base your inference. OK -- I was not intending to defend the frequentist position (I actually care little about bias), but rather simply explain the traditional justification for LS in terms of the G-M Theorem, which does not depend on a Gaussian error distribution. However, and this is really a subtle point, I still disagree that a Bayesian analysis reveals a hidden normality assumption for least squares. From a Bayesian and/or Info Theoretic POV, if you have only the first two moments, you are best off acting as if the distribution is normal. That is different from *assuming* that it is normal. If your analysis is based on a normality assumption, it is valid only to the extent that the normality assumption is valid. In contrast, a Bayesian is saying that it doesn't matter what the real distribution is, normal or not, you are best off using a normal distribution in your calculations, because a normal distribution assumes the least structure (or information) in your data given only the first two moments. Use of any other distribution would involve extra unwarranted assumptions.
