ALBERT

Note: Contents: Continuous Path Processes ; Continuous-Path Random Processes: Mathematical Prerequisites ; Basic Concepts and Examples in Finance ; Hitting Times: A Mix of Mathematics and Finance ; Complements on Brownian Motion ; Complements on Continuous Path Processes ; A Special Family of Diffusions: Bessel Processes ; Jump Processes ; Default Risk: An Enlargement of Filtration Approach ; Poisson Processes and Ruin Theory ; General Processes: Mathematical Facts ; Mixed Processes ; Lévy Processes

Note: Contents: 1 Introduction and examples. - 1.1 Introduction. - 1.2 Why Bayes?. - 1.2.1 Estimating the probability of a rare event. - 1.2.2 Building a predictive model. - 1.3 Where we are going. - 1.4 Discussion and further references. - 2 Belief, probability and exchangeability. - 2.1 Belief functions and probabilities. - 2.2 Events, partitions and Bayes' rule. - 2.3 Independence. - 2.4 Random variables. - 2.4.1 Discrete random variables. - 2.4.2 Continuous random variables. - 2.4.3 Descriptions of distributions. - 2.5 Joint distributions. - 2.6 Independent random variables. - 2.7 Exchangeability. - 2.8 de Finetti's theorem. - 2.9 Discussion and further references. - 3 One-parameter models. - 3.1 The binomial model. - 3.1.1 Inference for exchangeable binary data. - 3.1.2 Confidence regions. - 3.2 The Poisson model. - 3.2.1 Posterior inference . - 3.2.2 Example: Birth rates. - 3.3 Exponential families and conjugate priors. - 3.4 Discussion and further references. - 4 Monte Carlo approximation. - 4.1 The Monte Carlo method. - 4.2 Posterior inference for arbitrary functions. - 4.3 Sampling from predictive distributions. - 4.4 Posterior predictive model checking. - 4.5 Discussion and further references. - 5 The normal model. - 5.1 The normal model. - 5.2 Inference for the mean, conditional on the variance. - 5.3 Joint inference for the mean and variance. - 5.4 Bias, variance and mean squared error. - 5.5 Prior specification based on expectations. - 5.6 The normal model for non-normal data. - 5.7 Discussion and further references. - 6 Posterior approximation with the Gibbs sampler. - 6.1 A semiconjugate prior distribution. - 6.2 Discrete approximations. - 6.3 Sampling from the conditional distributions. - 6.4 Gibbs sampling. - 6.5 General properties of the Gibbs sampler. - 6.6 Introduction to MCMC diagnostics. - 6.7 Discussion and further references. - 7 The multivariate normal model. - 7.1 The multivariate normal density. - 7.2 A semiconjugate prior distribution for the mean. - 7.3 The inverse-Wishart distribution. - 7.4 Gibbs sampling of the mean and covariance. - 7.5 Missing data and imputation. - 7.6 Discussion and further references. - 8 Group comparisons and hierarchical modeling. - 8.1 Comparing two groups. - 8.2 Comparing multiple groups. - 8.2.1 Exchangeability and hierarchical models. - 8.3 The hierarchical normal model. - 8.3.1 Posterior inference. - 8.4 Example: Math scores in U.S. public schools. - 8.4.1 Prior distributions and posterior approximation. - 8.4.2 Posterior summaries and shrinkage. - 8.5 Hierarchical modeling of means and variances. - 8.5.1 Analysis of math score data. - 8.6 Discussion and further references. - 9 Linear regression. - 9.1 The linear regression model. - 9.1.1 Least squares estimation for the oxygen uptake data. - 9.2 Bayesian estimation for a regression model. - 9.2.1 A semiconjugate prior distribution. - 9.2.2 Default and weakly informative prior distributions. - 9.3 Model selection. - 9.3.1 Bayesian model comparison. - 9.3.2 Gibbs sampling and model averaging. - 9.4 Discussion and further references. - 10 Nonconjugate priors and Metropolis-Hastings algorithms. - 10.1 Generalized linear models. - 10.2 The Metropolis algorithm. - 10.3 The Metropolis algorithm for Poisson regression. - 10.4 Metropolis, Metropolis-Hastings and Gibbs. - 10.4.1 The Metropolis-Hastings algorithm. - 10.4.2 Why does the Metropolis-Hastings algorithm work?. - 10.5 Combining the Metropolis and Gibbs algorithms. - 10.5.1 A regression model with correlated errors. - 10.5.2 Analysis of the ice core data. -