Efron e Bayes

[Francisco Cribari <cribari@de.ufpe.br>]

Mesmo sob o risco de citar elogiosamente uma vez mais o Bradley Efron
(um dos estatísticos mais influentes da atualidade, sem dúvida), eu
gostaria de recomendar aos colegas a leitura de sua entrevista à
Statistical Science em 2003. O texto da entrevista em PDF (na íntegra)
pode ser obtido de 

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ss/1063994981 

Um extrato da entrevista (s/ o assunto que ocupou as últimas
mensagens): 

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   Tibshirani: In 1981, you asked the question: why isn't everyone a
Bayesian? Twenty-two years later do you think the proportion of
Bayesians among us has increased?
                                                           
   Efron: Yes, I think there's been a growth in the interest in Bayesian
statistics, particularly in England. The Royal Statistical Society seems
to feature Bayes in every issue. It's for good reasons. One of the good
reasons is that Bayesian statistics is different now than it was 20
years ago. It's more realistic and aims toward actually solving problems
instead of making philosophical points about why frequentism is wrong.
For example, José Bernardo just sent an e-mail announcing a conference
on the practical applications of Bayesian statistics. Of course, there's
been the computing revolution in Bayesian statistics.  The MCMC kind of
Gibbs sampling is a really impressive application. I believe that
empirical Bayes is the natural meeting ground between frequentism and
Bayesian theory, but that's the theory that hasn't boomed as I hoped it
would. The MCMC Bayes kind of theory has a drawback, in that it leads to
fairly simple priors being used because those are the ones that work
nicely for MCMC. As with all mathematical or computational advances,
people choose the path of least resistance. In some sense this conceals
the main problem of Bayesian statistics, choosing the prior. The nice
thing about empirical Bayes, when used correctly, is it finesses the
problem of choosing a prior on a high-dimensional parameter space, which
is the essence of the division between Bayesian and frequentist
statistics. 
 
   Morris: As regards empirical Bayes, you think it's an underachiever.
                                                          
   Efron: Empirical Bayes is an underachiever only in comparison with
things like the Wilcoxon test, which are used billions of times. People
do use empirical Bayes ideas or hierarchical modeling more generally,
but it really hasn't spread into the applications community. Also, I was
thinking in terms of the possible gains, and things like the Wilcoxon
test aren't really much of a gain over the t test in experienced hands.
But empirical Bayes can surprise even statisticians with how much you
can gain in practice. You can easily save 75% or 50% of the risk. So why
isn't it used more? The reason is that we're not too confident about the
theory and when it applies. Analysis of variance is incredibly useful:
it fits so many situations. Part of the reason it's so useful is because
Fisher taught scientists that statisticians can handle the analysis of
variance very well, so researchers designed experiments with us in mind.
If we get good at analyzing empirical Bayes situations and confident
about it both in the theoretical and applied sense, then I think
experimenters will start designing experiments that will make use of the
kind of parallel structure you need for empirical Bayes. Microarrays are
a good example of useful parallel structure.

   Holmes: Isn't there a problem of coherence when doing empirical
Bayes? What motivates mixing paradigms, taking a Bayesian viewpoint and
then using the data like a frequentist?

   Efron: The coherence question is a Bayesian's answer to optimality.
Optimality is what frequentists talk about. Bayesians counter with
coherence and say that frequentist theory tends to be incoherent in that
it doesn't combine information from different situations in a logical
way. And that's a perfectly good criticism, especially if you have to
combine some information. There are other attractive things about the
Bayesian approach. It's far more aggressively optimistic about modeling
than frequentism. Frequentism tends to be quite defensive, trying to
avoid making a statement that has a high probability of being wrong.
There's a lot I like about Bayesian statistics. What I don't like is
slapping on a prior and saying you've got an answer. It's very
dangerous, especially in high-dimensional problems. Bayesian theory is
quite impressive when you have a pretty good idea that the prior is at
least not harmful. You may have some complicated situation that
frequentism gets lost in, like multiple comparisons, and the Bayesian
approach then starts saying things that are interesting.

   Tibshirani: I think another important fact is that people tend to use
tools that give them answers when they didn't have any answer before.
Robust statistics was very big in the 1960s, but how much do we use it
now? Robust statistics gives a higher quality result in a situation
where we already had a result. The bootstrap gives an answer where we
had no answer before. That's the kind of tool people are going to use,
analysis of variance being a good example. It's a basic tool that gives
us an answer to questions that are important, scientifically.

   Efron: It helps that the bootstrap is easy to use and flexible. As
time goes by it gets easier and easier to apply. The kind of thing
that's hard to use is a theory like "uniform minimum variance unbiased,"
where you have to think of a new trick for each new case in order to
apply it. The things that go like gangbusters are ideas like maximum
likelihood estimation, where one algorithm fits all. So maybe what I was
trying to say was that empirical Bayes needs to be automated. 

   Tibshirani: I think the point that Brad is making is that a method
has to become semiautomatic before it's going to become widespread. If
it takes a Ph.D. in statistics to apply it every time, there just aren't
enough of us around to make it a widespread tool.

  Morris: So, for instance, lots of different packages now have methods,
for better or worse, that incorporate shrinkage of models. Has that gone
far enough, in your mind?

   Efron: When you use a Bayes or empirical Bayes estimate, you don't
have the safety net of each theta being estimated more or less
unbiasedly by its own "x," as you do in the classical theory. With
maximum likelihood estimation, each parameter is estimated in a way
that's fairly unbiased. If you use an empirical Bayes estimate,
everything gets pulled toward the central bulge. You have to grit your
teeth and believe the fact that even though any one estimate may be off,
overall you're getting a lot of improvement. That's what we have to get
people, including ourselves, to believe.

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Saudações, FC

Francisco Cribari-Neto               voice: +55-81-21267425
Departamento de Estatística          fax:   +55-81-21268422
Universidade Federal de Pernambuco   e-mail: cribari@de.ufpe.br
Recife/PE, 50740-540, Brazil         web: www.de.ufpe.br/~cribari/

    Behind every successful man stands an amazed mother-in-law.