Some time in 2006 (shortly after my ERROR06 conference), the trickle of irony and sometime flood of family feuds issuing from Bayesian forums drew me back into the Bayesian-frequentist debates.^{1 2} Suddenly sparks were flying, mostly kept shrouded within Bayesian walls, but nothing can long be kept secret even there. Spontaneous combustion is looming. The true-blue subjectivists were accusing the increasingly popular “objective” and “reference” Bayesians of practicing in bad faith; the new O-Bayesians (and frequentist-Bayesian unificationists) were taking pains to show they were not subjective; and some were calling the new Bayesian kids on the block “pseudo Bayesian.” Then there were the Bayesians somewhere in the middle (or perhaps out in left field) who, though they still use the Bayesian umbrella, were flatly denying the very idea that Bayesian updating fits anything they actually do in statistics.^{3} Obeisance to Bayesian reasoning remained, but on some kind of a priori philosophical grounds. Doesn’t the methodology used in practice really need a philosophy of its own? I say it does, and I want to provide this.

The result of my own interest here gave rise to a Kent-Virginia Tech workshop (in Kent)^{4} followed by the 2010 conference at the LSE, from which grew the special volume (see Mayo 2010, RMM volume, for examples and references).

Given it’s the middle of the night in London and I’ve just come off a day of trying to finalize Birnbaum, and put together a second paper on the philosophy of error statistics, my (still) jet-lagged mind is drifting (instead of sleeping), lured into a game of philosophical deconstruction of some of the recent claims the key players have been making.

Especially surprising to me, leaders of default Bayesianism, arguably the most predominant current form, began claiming that “violation of principles such as the likelihood principle is the price that has to be paid for objectivity” (Berger 2006, 394). As such, the default Bayesian may welcome with relief my critique of Birnbaum’s famous LP argument. (See my December 6 and 7 posts.) Even though “objectivity” is used very differently, there is still this odd sort of agreement in phrases uttered. While for us the violation is fully in order and is picked up on through the sampling distribution; for Bayesians it is anything but expected, and is picked up through model-dependent changes of priors (introducing strict incoherence).

It is noteworthy that default Bayesians don’t agree with each other even with respect to standard applications, as they readily admit. For instance, Bernardo, but not Berger, rejects the spiked prior that leads to pronounced conflicts between frequentist p-values and posteriors. While reonuncing the spikes makes the numbers agree (with frequentists), there is no evidence that the result is either an objective or rational degree of belief (as he intends) or an objective assessment of well-testedness (as our error statistician achieves). Bernardo wants to match the frequentist in the optional stopping case, but I take it Jim still adheres to the position of Berger and Wolpert 1988 an the SRP.

OK, so here’s an especially intriguing remark by Jim Berger that I think bears upon the current mindset. (Jim is aware of my efforts, it will come as no surprise that I’m sharing my meandering here.)

Too often I see people pretending to be subjectivists, and then using “weakly informative” priors that the objective Bayesian community knows are terrible and will give ridiculous answers; subjectivism is then being used as a shield to hide ignorance. . . . In my own more provocative moments, I claim that the only true subjectivists are the objective Bayesians, because they refuse to use subjectivism as a shield against criticism of sloppy pseudo-Bayesian practice. (Berger 2006, 463)

How might we deconstruct this fantastic remark of Berger’s?^{5} (Granted, this arises in his rejoinder to others, but this only heightens my interest in analyzing it.)

Here, “objective Bayesians” are understood as using (some scheme) of default or conventionally derived priors. One aspect of his remark is fairly clear: pseudo-Bayesian practice allows “terrible” priors to be used, and it would be better for them to appeal to conventional “default” priors that at least will not be so terrible (but in what respect?). It is the claim he makes in his “more provocative moments” that really invites deconstruction. Why would using the recommended conventional priors make them more like “true subjectivists”? I can think of several reasons—but none is really satisfactory, and all are (interestingly) perplexing. I am reminded of Sartre’s remarks in Being and Nothingness on bad faith and irony:

In irony a man annihilates what he posits within one and the same act; he leads us to believe in order not to be believed; he affirms to deny and denies to affirm; he creates a positive object but it has no being other than its nothingness.

So true! (Of course I am being ironic!) Back to teasing out what’s behind Berger’s remarks.

Now, it would seem that if she did use priors that correctly reflected her beliefs (call these priors “really informed by subjective opinions”(riso?), and that satisfied the Bayesian formal coherency requirements, then that would be defensible for a subjective Bayesian. But Berger notices that, in actuality, many Bayesians (the pseudo-Bayesians) do not use riso priors. Rather, they use various priors (the origin of which they’re unsure of) as if these really reflected their subjective judgments. In doing so, she (thinks that she) doesn’t have to justify them—she claims that they reflect subjective judgments (and who can argue with them?).

According to Berger here, the Bayesian community (except for the pseudo-Bayesians?) knows that they’re terrible, according to a shared criterion (is it non-Bayesian? Frequentist?). But I wonder: if, as far as the agent knows, these priors really do reflect the person’s beliefs, then would they still be terrible? It seems not. Or, if they still would be terrible, doesn’t that suggest a distinct criterion other than using “really informed” (as far as the agent knows) opinions or beliefs?

I await your interpretive efforts, and will post them (if received by the last day of Hanukkah)–Anonymous is OK.. ..

RMM 2011 refers to the special issue of the on-line journal, Rationality, Markets and Morals housing papers growing out of the LSE conference of June 2010: Statistical Science and Philosophy of Science: Where Do (Should) They Meet in 2011 and Beyond? http://www.rmm-journal.de/htdocs/st01.html

Follow-up blogposts to this call for U-Phils:

Contributed deconstructions of J. Berger:http://errorstatistics.com/2011/12/26/contributed-deconstructions-irony-bad-faith-3/

J. Berger on J. Berger:http://errorstatistics.com/2011/12/29/jim-berger-on-jim-berger/

[1] It was David Cox who first alerted me. Then there was Dongchu Sun, a statistician who visited at Virginia Tech.

[2] Yes, I’d given up on them, and was happy to spend all my remaining exiled days on philosophy of experiment.

[3] I’m not here including things like “Bayes nets,” which use conditional probability (as do we all) but are not really Bayesian.

[4] J. Williamson, J. Corfield and others at Kent co-hosted the first.

[5] As noted, Jim Berger is aware that I’m discussing this on my blog. I hope he will comment!

Berger, J. (2006),“The Case for Objective Bayesian Analysis”, and “Rejoinder”, *Bayesian Analysis* 1(3), 385–402; 457-464.

Mayo, D. (2011), “Statistical Science and Philosophy of Science: Where Do/Should They Meet in 2011 (and Beyond)?” RMM Vol. 2, 2011, 79–102

Sartre, J.P *Being and Nothingness: an essay in phenomenological ontology* (1943, Gallimard); English 1956, Philosophical Library Inc.

Senn, S. (2011) You may believe you are a Bayesian but you are probably wrong. Rationality, Markets and Morals RMM Vol. 2, 2011, 48–66

My guess is that there is a typo, and Berger meant to say “the only true objectivists are the objective

Bayesians…” in the quote above. Mystery solved! :)

I think that is exactly what he meant. It is highly subjective to decide to be flippant about priors.

This comment was published in Bayesian analysis which has an obviously specialist audience, the two articles and the comments on the two articles reveals a near unanimous preference for subjective Bayes as the foundations of statistics. To this narrow specialist audience “subjective” is a complement, an idealized limiting case of an optimal statistical analysis. If you have a philosophical objection to subjective Bayes (or Bayes in general) as the foundations of statistics then you are really far outside the target audience and understandably the comment will be opaque.

I think Berger is saying that an objective Bayesian might understand the consequences of diffuse priors better than a subjective Bayesian, he is probably employing both Bayesian and non-Bayesian criteria to investigate the consequence of priors, making objective Bayes a bit of a piece meal “theory”. My reading of the article is that Berger is a subjectivist, who is promoting tools outside standard subjective Bayesian theory (objective Bayes and frequentist) on practical grounds, it is interesting that the more extreme objective Bayes arguments of Jefreys and Jaynes seem to be largely abandoned now.

Of course the article reveals differences in Bayesians, but I think also reveals a remarkable convergence of opinion. Subjective Bayes is the foundations of statistics, but in an operational sense fully specifying subjective probabilities and then conditioning on observables is not remotely practical. Berger and Goldstein suggest different tools for dealing with this problem and the debate is largely carried out within this context (excluding Wasserman’s comments).

The differences of opinion that do exist in Bayesians may be used in order to identify weaknesses in the theory and be used in order to propomote a rival non-Bayesian foundation to statistics such as error statistics, which I think is the intention of the post…

Similarly there are differences of opinion in non-Bayesian statistics, and as you write the philosophy of error statistics seems isolated to a remote island. I appreciate you try to write in a whimsical style, but I have trouble working out if you are literally on Elba or not! My guess is that you are being both literal and metaphorical here, but please excuse me, I can’t always tell….

To my reading of the situation error statistics seems to differ radically from mainstream frequentist statistics in general in

a) thinking frequentist statisics has foundations

b) arguing that Birnbaum was wrong

In taking this position, I think you have staked out a really interesting part of the intellectual landscape for yourself, but I am sure a lonely one as well.

FWIW on both questions, my own opinion will be informed by how these arguments are digested by the scientific community at large… In particular for b) I am unwilling to try to get directly involved in the argument myself, due to limitations in my knowledge.

That is indeed an interesting and informative commentary. I am surprised but intrigued to know that Berger sees the subjective view as a strength. I find that hard to accept (subjectivism as a strength, that is). Also, I suspect only a certain subset of statisticians hold that subjective bayesianism is the foundation of statistics. Most scientists would not (I guess) and it would be interesting to know how many are familiar with the notion of subjective Bayesian inference at all. Likely not a substantial portion.

David: There’s much I’d want to say, but it will have to wait til I return—am in an airport, and no plugs in sight. However, frequentist error statistics is not intended to be different from “mainstream” frequentist statistics. Granted frequentists have been told, and some perhaps believe, they don’t have foundations (I say they do, I will come back to this, perhaps more explicitly than I have been, very soon). Granted, as well, they haven’t (to my knowledge) raised the particular critique of Birnbaum that I have , even though they have raised some (e.g., Cox, Durbin), and, most importantly, they reject the strong LP.

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