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Lecture notes are written by Elvis Cui and some materials are from scientific papers.
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Bayesian topics include:
- Bayesian conjugate linear models (models with conjugate prior).
- Connection between Bayesian and frequentist linear models.
- Sampling methods such as composition sampling, method of mixtures, MCMC algorithms, etc.
- Brook's lemma (aka Hammersley-Clifford theorem).
- Sherman-Woodbury-Morrison formula using multivariate statistics theory.
- Sequential Bayesian learning.
- Directed acyclic graph (DAG).
- Alternative ways to look at normal densities.
- Spatial models (CAR).
- Tensor product models, aka matrix-regression models.
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Classical topics include:
- Matrix theory with an emphasis on projection operators.
- Distribution theory, especially Fisher-Cochran's theorem.
- Least square theory: constrained estimation, conditions for a parameter to be estimable, etc.
- Multiple and partial correlation coefficients.
- Violation of assumptions in linear models and remedies.
- Hypothesis testing such as omnibus test, Fieller's theorem, Cook-Weiesberg's test, etc.
- Simultaneous inference such as Tukey's q, Scheffe's method, etc.
- Shrinkage and Bayes estimation (an unusual version of James-Stein estimator and its statistical properties are given).
- ANOVA mixed models.
- Linear mixed models with REML.
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Course materials are included.
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Homework are included (my own solutions).
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Linear models, Bayesian multivariate statistics, probabilistic machine leanring.
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