Tools for Statistical Inference

1. Introduction. - Exercises. - 2. Normal Approximations to Likelihoods and to Posteriors. - 2. 1. Likelihood/Posterior Density. - 2. 2. Specification of the Prior. - 2. 3. Maximum Likelihood. - 2. 4. Normal-Based Inference. - 2. 5. The ?-Method (Propagation of Errors). - 2. 6. Highest Posterior Density Regions. - Exercises. - 3. Nonnormal Approximations to Likelihoods and Posteriors. - 3. 1. Numerical Integration. - 3. 2. Posterior Moments and Marginalization Based on Laplace's Method. - 3. 3. Monte Carlo Methods. - Exercises. - 4. The EM Algorithm. - 4. 1. Introduction. - 4. 2. Theory. - 4. 3. EM in the Exponential Family. - 4. 4. Standard Errors in the Context of EM. - 4. 5. Monte Carlo Implementation of the E-Step. - 4. 6. Acceleration of EM (Louis' Turbo EM). - 4. 7. Facilitating the M-Step. - Exercises. - 5. The Data Augmentation Algorithm. - 5. 1. Introduction and Motivation. - 5. 2. Computing and Sampling from the Predictive Distribution. - 5. 3. Calculating the Content and Boundary of the HPD Region. - 5. 4. Remarks on the General Implementation of the Data Augmentation Algorithm. - 5. 5. Overview of the Convergence Theory of Data Augmentation. - 5. 6. Poor Man's Data Augmentation Algorithms. - 5. 7. Sampling/Importance Resampling (SIR). - 5. 8. General Imputation Methods. - 5. 9. Further Importance Sampling Ideas. - 5. 10. Sampling in the Context of Multinomial Data. - Exercises. - 6. Markov Chain Monte Carlo: The Gibbs Sampler and the Metropolis Algorithm. - 6. 1. Introduction to the Gibbs Sampler. - 6. 2. Examples. - 6. 3. Assessing Convergence of the Chain. - 6. 4. The Griddy Gibbs Sampler. - 6. 5. The Metropolis Algorithm. - 6. 6. Conditional Inference via the Gibbs Sampler. - Exercises. - References. Language: English