Posts Tagged With: frequentist-Bayesian unifications

Irony and Bad Faith: Deconstructing Bayesians-reblog

Posted on November 21, 2012 by Mayo

The recent post by Normal Deviate, and my comments on it, remind me of why/how I got back into the Bayesian-frequentist debates in 2006, as described in my first “deconstruction” (and “U-Phil”) on this blog (Dec 11, 2012):

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. Continue reading →

Categories: Likelihood Principle, objective Bayesians, Statistics | Tags: default priors, frequentist-Bayesian unifications, Jim Berger, objective Bayesians, pseudo-Bayesians

Matching Numbers Across Philosophies

Posted on April 25, 2012 by Mayo

The search for an agreement on numbers across different statistical philosophies is an understandable pastime in foundations of statistics. Perhaps identifying matching or unified numbers, apart from what they might mean, would offer a glimpse as to shared underlying goals? Jim Berger (2003) assures us there is no sacrilege in agreeing on methodology without philosophy, claiming “while the debate over interpretation can be strident, statistical practice is little affected as long as the reported numbers are the same” (Berger, 2003, p. 1).

Do readers agree?

Neyman and Pearson (or perhaps it was mostly Neyman) set out to determine when tests of statistical hypotheses may be considered “independent of probabilities a priori” ([p. 201). In such cases, frequentist and Bayesian may agree on a critical or rejection region.

The agreement between “default” Bayesians and frequentists in the case of one-sided Normal (IID) testing (known σ) is very familiar. As noted in Ghosh, Delampady, and Samanta (2006, p. 35), if we wish to reject a null value when “the posterior odds against it are 19:1 or more, i.e., if posterior probability of H₀ is < .05” then the rejection region matches that of the corresponding test of H₀, (at the .05 level) if that were the null hypothesis. By contrast, they go on to note the also familiar fact that there would be disagreement between the frequentist and Bayesian if one were instead testing the two sided: H₀: μ=μ₀ vs. H₁: μ≠μ₀ with known σ. In fact, the same outcome that would be regarded as evidence against the null in the one-sided test (for the default Bayesian and frequentist) can result in statistically significant results being construed as no evidence against the null —for the Bayesian– or even evidence for it (due to a spiked prior).[i] Continue reading →

Categories: Statistics | Tags: Bayesian probability, Frequentist inference, frequentist-Bayesian unifications, Statistical hypothesis testing

JIM BERGER ON JIM BERGER!

Posted on December 29, 2011 by Mayo

Fortunately, we have Jim Berger interpreting himself this evening (see December 11)

Jim Berger writes:

A few comments:

1. Objective Bayesian priors are often improper (i.e., have infinite total mass), but this is not a problem when they are developed correctly. But not every improper prior is satisfactory. For instance, the constant prior is known to be unsatisfactory in many situations. The ‘solution’ pseudo-Bayesians often use is to choose a constant prior over a large but bounded set (a ‘weakly informative’ prior), saying it is now proper and so all is well. This is not true; if the constant prior on the whole parameter space is bad, so will be the constant prior over the bounded set. The problem is, in part, that some people confuse proper priors with subjective priors and, having learned that true subjective priors are fine, incorrectly presume that weakly informative proper priors are fine. Continue reading →