Department of Statistics, University of Chicago
'
2014
$
An Overview of Objective Bayes Estimation
Jim Berger
Duke University and University of Chicago
Department of Statistics
University of Chicago,
5
Decision Theory
5.1
Notation
Here we introduce the basic notation.
2 : an unknown quantity aecting decision process.
a 2 A: action to be taken.
x 2 X : data
L(0 , a) : A ! R: loss for performing
8 The Likelihood Principle
Introduction
Consider parametric Bayesian inference. Since
( | x) = R
f (x | )()
,
f (x | )()d
It follows that if the prior () is fixed, the posterior ( | x) will be the sam
Bayesian Inference: Introduction
Preamble: Probability and Uncertainty
In Bayesian statistics, we use the calculus of probability to represent uncertainty.
Sometimes people (including possibly me!) wi
The Meaning of Probability
Acknowledgments
Some of these notes are based on ones provided to me by Peter Donnelly, which he used to
teach a similar class at the University of Chicago in the 1990s.
Sel
Simple Examples of Bayesian Statistics
The following examples are intended to give very simple illustrations of the use of Bayes
Theorem to compute posterior distributions from prior distributions. Th
Stat 30200: Mathematical Statistics 2
Recommended Reading
[1] J.O. Berger, Statistical decision theory and Bayesian analysis, Springer, 1985.
[2] J.O. Berger and T. Sellke, Testing a point null hypoth
Hierarchical Modeling and Exchangeability (see also Bernardo and Smith, Ch.
Note: these notes are work in progress, particularly the earlier material focussed on hierarchical modeling.
1
Hierarchical
3
Prior Distributions
There are several different approaches to specification of prior distributions. We can identify
at least three different types of specification:
1. Subjective specification: the
Multiple Testing
1
Multiple Testing
In practice, particularly in modern scientific applications, one is often faced with
the problem of testing not just one null hypothesis, but many.
Example 1: In br
4
4.1
Posterior Distributions
Posterior Summaries
In a Bayesian analysis, the posterior distribution encapsulates beliefs which have been updated in
the light of the data. Various natural summaries of