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preface
chapter 1: probability models
1.1 background
1.1.1 general concepts
1.1.2 classical statistics
1.1.3 bayesian statistics
1.2 exchangeability
1.2.1 distributional symmetry
1.2.2 frequency and exchangeability
1.3 parametric models
1.3.1 prior, posterior, and predictive distributions
1.3.2 improper prior distributions
1.3.3 choosing probability distributions
1.4 definetti's representation theorem
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preface
chapter 1: probability models
1.1 background
1.1.1 general concepts
1.1.2 classical statistics
1.1.3 bayesian statistics
1.2 exchangeability
1.2.1 distributional symmetry
1.2.2 frequency and exchangeability
1.3 parametric models
1.3.1 prior, posterior, and predictive distributions
1.3.2 improper prior distributions
1.3.3 choosing probability distributions
1.4 definetti's representation theorem
1.4.1 understanding the theorems
1.4.2 the mathematical statements
1.4.3 some examples
1.5 proofs of definetti's theorem and related results*
1.5.1 strong law of large numbers
.1.5.2 the bernoulli case
1.5.3 the general finite case'
1.5.4 the general infinite case
1.5.5 formal introduction to parametric models*
1.6 infinite-dimensional parameters*
1.6.1 dirichlet processes
1.6.2 tailfree processes+
1.7 problems
chapter 2: sufficient statistics
2.1 definitions
2.1.1 notational overview
2.1.2 sufficiency
2.1.3 minimal and complete sufficiency
2.1.4 ancillarity
2.2 exponential families of distributions
2.2.1 basic properties
2.2.2 smoothness properties
2.2.3 a characterization theorem*
2.3 information
2.3.1 fisher information
2.3.2 kullback-leibler information
2.3.3 conditional information*
2.3.4 jeffreys' prior*
2.4 extremal families'
2.4.1 the main results
2.4.2 examples
2.4.3 proofs+
2.5 problems
chapter 3: decision theory
3.1decision problems
3.1.1 framework
3.1.2 elements of bayesian decision theory
3.1.3 elements of classical decision theory
3.1.4 summary
3.2 classical decision theory
3.2.1 the role of sufficient statistics
3.2.2 admissibility
3.2.3 james-stein estimators
3.2.4 minimax rules
3.2.5 complete classes
3.3 axiomatic derivation of decision theory'
3.3.1 definitions and axioms
3.3.2 examples
3.3.3 the main theorems
3.3.4 relation to decision theory
3.3.5 proofs of the main theorems'
3.3.6 state-dependent utility*
3.4 problems:
chapter 4: hypothesis testing
4.1 introduction
4.1.1 a special kind of decision problem
4.1.2 pure significance tests
4.2 bayesian solutions
4.2.1 testing in general
4.2.2 bayes factors
4.3 most powerful tests
4.3.1 simple hypotheses and alternatives
4.3.2 simple hypotheses, composite alternatives
4.3.3 one-sided tests
4.3.4 two-sided hypotheses
4.4 unbiased tests
4.4. i general results
4.4.2 interval hypotheses
4.4.3 point hypotheses
4.5 nuisance parameters
4.5.1 neyman structure
4.5.2 tests about natural parameters
4.5.3 linear combinations of natural parameters
4.5.4 other two-sided cases'
4.5.5 likelihood ratio tests
4.5.6 the standard f-test as a bayes rule*.
4.6 p-values
4.6.1 definitions and examples
4.6.2 p-values and bayes factors
4.7 problems
chapter 5: estimation
5.1 point estimation
5.1.1 minimum variance unbiased estimation
5.1.2 lower bounds on the variance of unbiased estimators
5.1.3 maximum likelihood estimation
5.1.4 bayesian estimation
5.1.5 robust estimation*
5.2 set estimation
5.2.1 confidence sets
5.2.2 prediction sets*
5.2.3 tolerance sets*
5.2.4 bayesian set estimation
5.2.5 decision theoretic set estimation'
5.3 the bootstrap*
5.3.1 the general concept
5.3.2 standard deviations and bias
5.3.3 bootstrap confidence intervals
5.4 problems
chapter 6: equivariance
6.1 common examples
6.1.1 location problems
6.1.2 scale problems'
6.2 equivariant decision theory
6.2.1 groups of transformations
6.2.2 equivariance and changes of units
6.2.3 minimum risk equivariant decisions
6.3 testing and confidence intervals'
6.3.1 p-values in invariant problems
6.3.2 equivariant confidence sets
6.3.3 invariant tests*
6.4 problems
chapter 7: large sample theory
7.1 convergence concepts
7.1.1 deterministic convergence
7.1.2 stochastic convergence
7.1.3 the delta method
7.2 sample quantiles
7.2.1 a single quantile
7.2.2 several quantiles
7.2.3 linear combinations of quantiles'
7.3 large sample estimation
7.3.1 some principles of large sample estimation
7.3.2 maximum likelihood estimators
7.3.3 mles in exponential families
7.3.4 examples of inconsistent mles
7.3.5 asymptotic normality of mles
7.3.6 asymptotic properties of m-estimators'
7.4 large sample properties of posterior distributions
7.4.1 consistency of posterior distributions+
7.4.2 asymptotic normality of posterior distributions
7.4.3 laplace approximations to posterior distributions*
7.4.4 asymptotic agreement of predictive distributions+
7.5 large sample tests
7.5.1 likelihood ratio tests
7.5.2 chi-squared goodness of fit tests
7.6 problems
chapter 8: hierarchical models
8.1 introduction
8.1.1 general hierarchical models
8.1.2 partial exchangeability'
8.1.3 examples of the representation theorem'
8.2 normal linear models
8.2.1 one-way anova
8.2.2 two-way mixed model anova'
8.2.3 hypothesis testing
8.3 nonnormal models'
8.3.1 poisson process data
8.3.2 bernoulli process data
8.4 empirical bayes analysis*
8.4.1 nayve empirical bayes
8.4.2 adjusted empirical bayes
8.4.3 unequal variance case
8.5 successive substitution sampling
8.5.1 the general algorithm
8.5.2 normal hierarchical models
8.5.3 nonnormal models
8.6 mixtures of models
8.6.1 general mixture models
8.6.2 outliers
8.6.3 bayesian robustness
8.7 problems
chapter 9: sequential analysis
9.1 sequential decision problems
9.2 the sequential probability ratio test
9.3 interval estimation*
9.4 the relevance of stopping rules
9.5 problems
appendix a: measure and integration theory
a.1 overview
a.1.1 definitions
a.1.2 measurable functions
a.1.3 integration
a.1.4 absolute continuity
a.2 measures
a.3 measurable functions
a.4 integration
a.5 product spaces
a.6 absolute continuity
a.7 problems
appendix b: probability theory
b.1 overview
b.i.1 mathematical probability
b.l.2 conditioning
b.1.3 limit theorems
b.2 mathematical probability
b.2.1 random quantities and distributions
b.2.2 some useful inequalities
b.3 conditioning
b.3.1 conditional expectations
b.3.2 borel spaces'
b.3.3 conditional densities
b.3.4 conditional independence
b.3.5 the law of total probability
b.4 limit theorems
b.4.1 convergence in distribution and in probability
b.4.2 characteristic functions
b.5 stochastic processes
b.5.1 introduction
b.5.2 martingales+
b.5.3 markov chains*
b.5.4 general stochastic processes
b.6 subjective probability
b.7 simulation*
b.8 problems
appendix c: mathematical theorems not proven here
c.1 real analysis
c.2 complex analysis
c.3 functional analysis
appendix d: summary of distributions
d.1 univariate continuous distributions
d.2 univariate discrete distributions
d.3 multivariate distributions
references
notation and abbreviation index
name index
subject index