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  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
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\begin{center}
  4:15 p.m., Wednesday, February 1, 2006\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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\begin{center}
  \textsl{Francisco J. Samaniego\\
    University of California, Davis}

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  \textbf{On Conjugacy and Self-Consistency in Bayesian Inference}
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\noindent
The notion of self-consistency of inferential procedures has
arisen in a number of statistical contexts. In a survival analysis
setting, Efron (Proc. Berkeley Symp., 1967) defined self consistency
in terms of a natural recursive relationship that nonparametric
estimators might satisfy, and showed that the Kaplan-Meier estimator
was the unique self-consistent estimator of the survival function on
the interval containing all deaths and censoring times. Tsai and
Crowley (Ann. Stat., 1985) identified self –consistent estimators as
the unique fixed points of nonparametric EM algorithms. In this talk,
self consistency is defined in the context of Bayes estimation,
relative to squared error loss, of a parameter theta of an exponential
family of distributions. We define a prior distribution (or Bayes
estimator) as self consistent if the equation $E(\theta | T =
E(\theta))) = E(\theta)$ is satisfied, where T is a sufficient and
unbiased estimator of $\theta$ (or the UMVUE of $\theta$ under mild
additional assumptions). This equation simply states that, if your
experimental outcome agrees with your prior opinion about the
parameter, then the experiment should not change your opinion about
$\theta$. While this condition seems natural and compelling, many
prior distributions, including both “objective” and proper priors, do
not enjoy this property. Some characterization results for families of
self-consistent priors are obtained. The concept of conjugacy will be
broadened by relaxing the exponentiality condition imposed on the
prior model by Diaconis and Ylvisaker (Ann. Stat., 1979) while
retaining their essential condition of a linear posterior mean. I'll
conclude by specifying when Bayes estimators relative to priors
belonging to a broad conjugate and self-consistent family outperform
classical procedures (in the sense of Samaniego and Reneau (JASA,
1994)).

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