Comparing Bayesian and Frequentist Estimators of a Scalar Parameter
As should be evident from the discussion in the three preceding chapters, both the Bayesian and the frequentist approaches to estimation have positive attributes, and yet both also have vulnerabilities that can lead to poor and misleading inferences. The Bayesian paradigm appears to have the advantage in terms of pure logic, both in its foundations and in the methodology that’s built upon them.We have noted, however, that a logically consistent analysis might rightly be judged to be inadequate when it leads to a conclusion that is off the mark. The Frequentist school, on the other hand, has an apparent edge in terms of the notion of “objectivity,” as it proceeds on the basis of a data-driven model and does not utilize “subjective” inputs concerning unknown population parameters whose influence is often difficult to identify and may, in some circumstances, be detrimental. But “objectivity” has been seen to be a two-edged sword, as simple examples make it abundantly clear that subjective inputs can, at times, save an analyst from disaster. Our examination of asymptotic methods in Statistics leads to the conclusion that, under reasonably broad conditions, the two theories of estimation result in solutions that may be described as equivalent (albeit with respect to a frequentist measure of merit). Ease of application has been discussed, and while it is hardly a criterion one would want to place undue weight on when choosing an approach in any serious application, the issue does help us understand why frequentist methods might be the more popular options in certain kinds of applications. In modern computing environments, Bayesian analyses are now feasible in a wide range of models and problems, and the “ease of application” issue might well be considered a draw at this point in time.
KeywordsPrior Distribution Unbiased Estimator Frequentist Estimator Exponential Family Error Loss
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