Friday, June 6, 2014

Frequentist vs. Bayesian Analysis

"Statisticians should readily use both Bayesian and frequentist ideas."

So begins a 2004 paper by Bayarri and Berger, "The Interplay of Bayesian and Frequentist Analysis", Statistical Science, 19(1), 58-80.

Let's re-phrase that opening sentence: "Econometricians should readily use both Bayesian and frequentist ideas."

Before turning to economics, my undergraduate training was in statistics and pure mathematics. My statistical training (in the 1960's) came from professors who were staunchly Bayesian - at a time when it was definitely "them and us". With few exceptions, the attitude was that "if you're not with us, then you're against us". And this was true on both sides of the Frequentist-Bayesian divide.

Hardly a healthy situation - but we've seen similar philosophical divisions throughout the history of economics, and in pretty much every other discipline at some point.

After a very orthodox training in econometrics (based largely on the texts of Johnston, and Malinvaud) I ended up doing my Ph.D. dissertation on some problems in Bayesian econometrics - supervised by a wonderful man who probably didn't have a Bayesian bone in his body. My first J. Econometrics paper looked at some of the sampling properties of certain Bayes estimators. How non-Bayesian can you get?

So, I've always told students that they need to be flexible in their econometric thinking, and they need to be prepared to use both frequentist and Bayesian tools. Time has proved me right, I believe. Modern econometric practice takes advantage of a healthy mix of ideas and techniques drawn from both tool boxes.

Yes, this has been made possible by the considerable advances that we have seen in computing methods and power in recent decades. But it's also reflected something of a shift in the mind-set of statisticians and econometricians alike.

Here's the concluding section of the Bayarri and Berger paper, in its entirety (pp.77-78):
"It seems quite clear that both Bayesian and frequentist philosophy are here to stay, and that we should not expect either to disappear in the future. This is not to say that all Bayesian or all frequentist methodology is fine and will survive. To the contrary, there are many areas of frequentist methodology that should be replaced by (existing) Bayesian methodology that provides superior answers, and the verdict is still out on those Bayesian methodologies that have been exposed as having potentially serious frequentist problems. 
Philosophical unification of the Bayesian and frequentist positions is not likely, nor desirable, since each illuminates a different aspect of statistical inference. We can hope, however, that we will eventually have a general methodological unification, with both Bayesian and frequentists agreeing on a body of standard statistical procedures for general use"
I hope that student followers of this blog will take the time to read the Bayarri and Berger paper, and to learn more about Bayesian methods.

© 2014, David E. Giles

2 comments:

  1. [I believe it's Bayarri.] It's fine to say we need both, that “philosophical unification of the Bayesian and frequentist positions is not likely, nor desirable, since each illuminates a different aspect of statistical inference", the worry I find is that Bayesians may not be open to seeing the distinct frequentist aspect of statistical inference. Indeed, the very idea of their being distinct functions, each valuable, is in tension with the claim that "there are many areas of frequentist methodology that should be replaced by (existing) Bayesian methodology that provides superior answers, and the verdict is still out on those Bayesian methodologies that have been exposed as having potentially serious frequentist problems." A judgment of "superior answers" and recommending replacement (!) is to assume one standpoint of appraisal (the Bayesian). Note too how Bayesian answers with "potentially serious frequentist problems" are allowed to have the verdict be "still out". Whether one wants to use the terms frequentist (or sampling theory, or error probability statistics or the like), these methods have demonstrated, over some time now, their abilities to control and assess error rates of methods. If that is to be understood as a distinct goal from Bayesian methods, that’s fine, but then the claims to superiority (without qualification) are unwarranted. (Some of the lessons about the importance of error control are perhaps being relearned nowadays in the face of reproducibility problems.) But my real point is simply to highlight a certain bias in typical claims about pluralism of foundations. Possibly it cannot be helped.

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  2. This is reminiscent of the hard-headed attitude of R. W. Hamming, who had no patience with any philosophy of statistics if it would prevent him from adopting the best method to solve the problem in hand.

    Gelman and Shalizi take a different and more subtle view in their paper "Philosophy and the practice of Bayesian statistics". They come very close to being Bayesian modelers with a frequentist philosophy, in that they regard the test of a model to be its frequentist properties - and they are clear that every model requires testing. In their more figurative flights, models have nearly an adversarial relationship with the modeler, almost anthropomorphized as agents requiring constant supervision.

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