Ebook: Conditional Independence in Applied Probability

A. Preliminaries.- 1. Probability Spaces and Random Vectors.- 2. Mathematical Expectation.- 3. Problems.- B. Conditional Independence of Events.- 1. The Concept.- 2. Some Patterns of Probable Inference.- 3. A Classification Problem.- 4. Problems.- C. Conditional Expectation.- 1. Conditioning by an Event.- 2. Conditioning by a Random Vector-Special Cases.- 3. Conditioning by a Random Vector-General Case.- 4. Properties of Conditional Expectation.- 5. Conditional Distributions.- 6. Conditional Distributions and Bayes’ Theorem.- 7. Proofs of Properties of Conditional Expectation.- 8. Problems.- D. Conditional Independence, Given a Random Vector.- 1. The Concept and Some Basic Properties.- 2. Some Elements of Bayesian Analysis.- 3. A One-Stage Bayesian Decision Model.- 4. A Dynamic-Programming Example.- 5. Proofs of the Basic Properties.- 6. Problems.- E. Markov Processes and Conditional Independence.- 1. Discrete-Parameter Markov Processes.- 2. Markov Chains with Costs and Rewards.- 3. Continuous-Parameter Markov Processes.- 4. The Chapman-Kolmogorov Equation.- 5. Proof of a Basic Theorem on Markov Processes.- 6. Problems.- Appendices.- Appendix I. Properties of Mathematical Expectation.- Appendix II. Properties of Conditional Expectation, Given a Random Vector.- Appendix III. Properties of Conditional Independence, Given a Random Vector.- References.- Selected Answers, Hints, and Key Steps.