I very much appreciate C. Robert and A. Jaffe sharing some reflections on Stephen Senn’s article for this blog, especially as I have only met these two statisticians recently, at different conferences. My only wish is that they had taken a bit more seriously my request to “hold (a portion of) the text at ‘arm’s length,’ as it were. Cycle around it, slowly. Give it a generous interpretation, then cycle around it again self-critically” (January 13, 2011). (I conceded it would feel foreign, but I strongly recommend it!)

Since these authors have given bloglinks, I’ll just note them here and give a few brief responses:

**Christian Robert**

**Mayo’s brief remarks**:

As I see it, Robert overlooks the most difficult challenge Senn raises–namely that in practice, people who claim to have carried out a (subjective) Bayesian analysis have actually done something very different—but that then they heap credit on the Bayesian ideal (what I called the “grace and amen” routine). He instead attempts to take the convenient escape route I warned of in my post on Senn (Jan 15[i]); namely to insist that they are still approximating the subjective Bayesian way, as if, examples to the contrary, Senn is merely pointing up some minor imperfections rather than a relinquishing of what are still regarded as core principles of subjective Bayesianism.

For instance, when default Bayesian admit to violating long-held principles, they seem not merely to be making concessions to human imperfections but rather denying fundamental principles and assumptions still regarded as integral to subjective Bayesianism. e.g., the likelihood principle, Dutch Book arguments, and even that inductive inference follows Bayes’s theorem. These foundational problems demand a greater gesture than a minor compromise or admission that no one’s perfect!

Robert’s claim that Senn “freezes the (Bayesian reasoning about the) Bayesian paradigm in its de Finetti phase-state” equally cuts no ice; Senn’s examples are taken from recent Bayesian work, and the onus is on Robert, a subjective Bayesian, to show that Senn’s criticisms do not stand.

As I keep asking, doesn’t what is actually doing the work deserve its own epistemological grounding? I say it does.

As I keep asking, doesn’t what is actually doing the work deserve its own epistemological grounding? I say it does.

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**Andrew Jaffe**

**Mayo’s brief remarks**:

Andrew Jaffe protests, ‘no no Senn doesn’t understand how we really function in practice…it’s not at all like that’ (“I think that these criticisms mis-state the practice of Bayesian statistics, at least by the scientists I know (mostly cosmologists and astronomers).” But that’s precisely what Senn does understand and why he faults the Bayesians for thanking the subjective Bayesian paradigmfor what we (readers) are about to receive, alleging that they owe it all to Bayes. No credit properly goes back to Bayesian ways if they are not responsible for the touted results.

Says Jaffe, “Rather, most of us take a vaguely Jaynesian view, after the cranky Edwin Jaynes” Jaffe says. Well I agree that Jaynes was cranky. This is the objective Bayesian I cited in an earlier post (Dec. 6, 2011) as declaring (cantankerously) that outcomes other than the one observed can’t matter to inference:

“The question of how often a given situation would arise is utterly irrelevant to the question how we should reason when it does arise. I don’t know how many times this simple fact will have to be pointed out before statisticians of ‘frequentist” persuasions will take note of it.” (Jaynes 1976, 247)

To which I replied:

“What we wonder is how many times we will have to point out that to us, reasoning from the result that arose is crucially dependent on how often it would have arisen…..”

I just don’t see how a subjective Bayesian can be at all comforted by Jaffe’s reference to a free and easy spirit type of Bayes, much less when he adds: “This is a point of view espoused most forcefully by Andrew Gelman” given that Gelman has pretty clearly denounced the very idea of doing inductive inference by way of Bayes’s theorem.[ii]

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[i] I had written, “But now that nearly no Bayesians explicitly advocate the one true subjective Bayesian ideal, more is needed. Their position has shifted. While adhering to the BADD ideal, they will still describe their methods as mere approximations of that ideal. After all, they will (and do) claim they can’t be perfect,but the Bayesian ideal still lights the way, and therefore discredits all Senn’s-ible criticism of their claim that all you need is Bayes.”

[ii] I now have Gelman’s to add, on the next post.

FWIW, I’m learning how to model metabolisms, and when modelling biological systems, inductive inference by way of Bayes is pretty much the only way of doing it.

Bayes’s theorem is just that, a theorem from the defn of conditional probability. The big issue, at least one of them, is how do you define and where do you get the priors? The mere use of conditional probability is something everyone does

insofar as probability is used and doesn’t make one a Bayesian. Same for Bayes’s nets. So are there frequentist priors in your case or…?

Oh, priors is actually my biggest bugbear. As far as I can tell, there are two possibilities:

1. There is a SuperSekrit Magic Realm of frequentist priors which other biologists can access but which I do not yet have the keys to;

2. There is nothing; you just Make Something Up.

Sorry. More seriously: we always start from incomplete knowledge and limited data wrt most paramaters, though cumulative experience can give us a guess at the nature of certain interactions — but proceeding from there, the standard path is the Jaynes/D’Agostini one. Given the nature of biological systems and their feedbacks, and the fact that at a cellular level the system is frankly full of stochastic noise, we can’t ever afford to assume that an unknown parameter is not a random variate. It almost always ends up being probability distributions. And I was under the impression (correct me if this is wrong, please?) that that’s not really what frequentism is about.

Be aware, I’m still learning this myself; don’t ever count me as an expert.