Principles of Uncertainty

Probability
Avoiding being a sure loser
Disjoint events
Events not necessarily disjoint
Random variables, also known as uncertain quantities
Finite number of values
Other properties of expectation
Coherence implies not a sure loser
Expectations and limits

Conditional Probability and Bayes Theorem
Conditional probability
The Birthday Problem
Simpson's Paradox
Bayes Theorem
Independence of events
The Monty Hall problem
Gambler's Ruin problem
Iterated Expectations and Independence
The binomial and multinomial distributions
Sampling without replacement
Variance and covariance
A short introduction to multivariate thinking
Tchebychev's inequality

Discrete Random Variables
Countably many possible values
Finite additivity
Countable Additivity
Properties of countable additivity
Dynamic sure loss

Probability generating functions
Geometric random variables
The negative binomial random variable
The Poisson random variable
Cumulative distribution function
Dominated and bounded convergence

Continuous Random Variables
Introduction
Joint distributions
Conditional distributions and independence
Existence and properties of expectations
Extensions
An interesting relationship between cdf's and expectations of continuous random variables
Chapter retrospective so far
Bounded and dominated convergence
The Riemann-Stieltjes integral
The McShane-Stieltjes Integral
The road from here
The strong law of large numbers

Transformations
Introduction
Discrete Random Variables
Univariate Continuous Distributions
Linear spaces
Permutations
Number systems; DeMoivre's formula
Determinants
Eigenvalues, eigenvectors and decompositions
Non-linear transformations
The Borel-Kolmogorov paradox

Normal Distribution
Introduction
Moment generating functions
Characteristic functions
Trigonometric Polynomials
A Weierstrass approximation theorem
Uniqueness of characteristic functions
Characteristic function and moments
Continuity Theorem
The Normal distribution
Multivariate normal distributions
Limit theorems

Making Decisions
Introduction
An example
In greater generality
The St. Petersburg Paradox
Risk aversion
Log (fortune) as utility
Decisions after seeing data
The expected value of sample information
An example
Randomized decisions
Sequential decisions

Conjugate Analysis
A simple normal-normal case
A multivariate normal case, known precision
The normal linear model with known precision
The gamma distribution
Uncertain Mean and Precision
The normal linear model, uncertain precision
The Wishart distribution
Both mean and precision matrix uncertain
The beta and Dirichlet distributions
The exponential family
Large sample theory for Bayesians
Some general perspective

Hierarchical Structuring of a Model
Introduction
Missing data
Meta-analysis
Model uncertainty/model choice
Graphical Hierarchical Models
Causation

Markov Chain Monte Carlo
Introduction
Simulation
The Metropolis Hasting Algorithm
Extensions and special cases
Practical considerations
Variable dimensions: Reversible jumps

Multiparty Problems
A simple three-stage game
Private information
Design for another's analysis
Optimal Bayesian Randomization
Simultaneous moves
The Allais and Ellsberg paradoxes
Forming a Bayesian group

Exploration of Old Ideas
Introduction
Testing
Confidence intervals and sets
Estimation
Choosing among models
Goodness of fit
Sampling theory statistics
Objective" Bayesian Methods

Epilogue: Applications
Computation
A final thought