a Bayesian interpretation of FDRs?

This week, I happened to re-read John Storey’ 2003 “The positive discovery rate: a Bayesian interpretation and the q-value”, because I wanted to check a connection with our testing by mixture [still in limbo] paper. I however failed to find what I was looking for because I could not find any Bayesian flavour in the paper apart from an FRD expressed as a “posterior probability” of the null, in the sense that the setting was one of opposing two simple hypotheses. When there is an unknown parameter common to the multiple hypotheses being tested, a prior distribution on the parameter makes these multiple hypotheses connected. What makes the connection puzzling is the assumption that the observed statistics defining the significance region are independent (Theorem 1). And it seems to depend on the choice of the significance region, which should be induced by the Bayesian modelling, not the opposite. (This alternative explanation does not help either, maybe because it is on baseball… Or maybe because the sentence “If a player’s [posterior mean] is above .3, it’s more likely than not that their true average is as well” does not seem to appear naturally from a Bayesian formulation.) [Disclaimer: I am not hinting at anything wrong or objectionable in Storey’s paper, just being puzzled by the Bayesian tag!]

same risk, different estimators

An interesting question on X validated reminded me of the epiphany I had some twenty years ago when reading a Annals of Statistics paper by Anirban Das Gupta and Bill Strawderman on shrinkage estimators, namely that some estimators shared the same risk function, meaning their integrated loss was the same for all values of the parameter. As indicated in this question, Stefan‘s instructor seems to believe that two estimators having the same risk function must be a.s. identical. Which is not true as exemplified by the James-Stein (1960) estimator with scale 2(p-2), which has constant risk p, just like the maximum likelihood estimator. I presume the confusion stemmed from the concept of completeness, where having a function with constant expectation under all values of the parameter implies that this function is constant. But, for loss functions, the concept does not apply since the loss depends both on the observation (that is complete in a Normal model) and on the parameter.

Charles M. Stein [1920-2016]

I have just heard that Charles Stein, Professor at Stanford University, passed away last night. Although the following image is definitely over-used, I truly feel this is the departure of a giant of statistics.  He has been deeply influential on the fields of probability and mathematical statistics, primarily in decision theory and approximation techniques. On the first field, he led to considerable changes in the perception of optimality by exhibiting the Stein phenomenon, where the aggregation of several admissible estimators of unrelated quantities may (and will) become inadmissible for the joint estimation of those quantities! Although the result can be explained by mathematical and statistical reasoning, it was still dubbed a paradox due to its counter-intuitive nature. More foundationally, it led to expose the ill-posed nature of frequentist optimality criteria and certainly contributed to the Bayesian renewal of the 1980’s, before the MCMC revolution. (It definitely contributed to my own move, as I started working on the Stein phenomenon during my thesis, before realising the fundamentally Bayesian nature of the domination results.)

“…the Bayesian point of view is often accompanied by an insistence that people ought to agree to a certain doctrine even without really knowing what this doctrine is.” (Statistical Science, 1986)

The second major contribution of Charles Stein was the introduction of a new technique for normal approximation that is now called the Stein method. It relies on a differential operator and produces estimates of approximation error in Central Limit theorems, even in dependent settings. While I am much less familiar with this aspect of Charles Stein’s work, I believe the impact it has had on the field is much more profound and durable than the Stein effect in Normal mean estimation.

(During the Vietnam War, he was quite active in the anti-war movement and the above picture from 2003 shows that his opinions had not shifted over time!) A giant truly has gone.

Bayesian propaganda?

“The question is about frequentist approach. Bayesian is admissable [sic] only by wrong definition as it starts with the assumption that the prior is the correct pre-information. James-Stein beats OLS without assumptions. If there is an admissable [sic] frequentist estimator then it will correspond to a true objective prior.”

I had a wee bit of a (minor, very minor!) communication problem on X validated, about a question on the existence of admissible estimators of the linear regression coefficient in multiple dimensions, under squared error loss. When I first replied that all Bayes estimators with finite risk were de facto admissible, I got the above reply, which clearly misses the point, and as I had edited the OP question to include more tags, the edited version was reverted with a comment about Bayesian propaganda! This is rather funny, if not hilarious, as (a) Bayes estimators are indeed admissible in the classical or frequentist sense—I actually fail to see a definition of admissibility in the Bayesian sense—and (b) the complete class theorems of Wald, Stein, and others (like Jack Kiefer, Larry Brown, and Jim Berger) come from the frequentist quest for best estimator(s). To make my point clearer, I also reproduced in my answer the Stein’s necessary and sufficient condition for admissibility from my book but it did not help, as the theorem was “too complex for [the OP] to understand”, which shows in fine the point of reading textbooks!

Cancún, ISBA 2014 [day #1]

The first full day of talks at ISBA 2014, Cancún, was full of goodies, from the three early talks on specifically developed software, including one by Daniel Lee on STAN that completed the one given by Bob Carpenter a few weeks ago in Paris (which gives me the opportunity to advertise STAN tee-shirts!). To the poster session (which just started a wee bit late for my conference sleep pattern!). Sylvia Richardson gave an impressive lecture full of information on Bayesian genomics. I also enjoyed very much two sessions with young Bayesian statisticians, one on Bayesian econometrics and the other one more diverse and sponsored by ISBA. Overall, and this also applies to the programme of the following days, I found that the proportion of non-parametric talks was quite high this year, possibly signalling a switch in the community and the interest of Bayesians. And conversely very few talks on computing related issues. (With most scheduled after my early departure…)

In the first of those sessions, Brendan Kline talked about partially identified parameters, a topic quite close to my interests, although I did not buy the overall modelling adopted in the analysis. For instance, Brendan Kline presented the example of a parameter θ that is the expectation of a random variable Y which is indirectly observed through x <Y< x̅ . While he maintained that inference should be restricted to an interval around θ and that using a prior on θ was doomed to fail (and against econometrics culture), I would have prefered to see this example as a missing data one, with both x and x̅ containing information about θ. And somewhat object to the argument against the prior as it would equally apply to any prior modelling. Although unrelated in the themes, Angela Bitto presented a work on the impact of different prior modellings on the estimation of time-varying parameters in time-series models. À la Harrison and West 1994 Discriminating between good and poor shrinkage in a way I could not spot. Unless it was based on the data fit (horror!). And a third talk of interest by Andriy Norets that (very loosely) related to Angela’s talk by presenting a framework to modify credible sets towards frequentist properties: one example was the credible interval on a positive normal mean that led to a frequency-valid confidence interval with a modified prior. This reminded me very much of the shrinkage confidence intervals of the James-Stein era.