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