Prologue
I Talking Probability
1. 1 The Eminence of Models
1.A For brave eyes only
2. Building Vocabulary
2.1. Probability
2.1.1 Basic rules
2.2. Conditional probability
2.2.1 Independence
2.2.2 Law of total probability
2.2.3 Bayes
law
2.3. Random variables
2.3.1 Summation and integration
2.3.2 Expectations and variances
2.3.3 Two simple distributions
2.4. The bell curve
3. Gaining Fluency
3.1. Multiple random quantities
3.1.1 Higher-dimensional problems
3.2. Two <"hard>" problems
3.2.1
Functions of random variables
3.2.2 Compound distributions
3A. Sums of independent random variables
3.A.1 Convolutions
3.A.2 Moment generating functions
3.A.3 Formulae for expectations and variances
II Doing Statistics
4. An Overview of Statistics
4.1.
Frequentist approach
4.1.1 Functions of random variables
4.2. Bayesian approach
4.2.1 Compound distributions
4.3. Two more distributions
4.3.1 Poisson distribution
4.3.2 Gamma distribution
4.A. Expectation and variance of the Poisson
4.B. Waiting time in Poisson
process
5. The Frequentist Approach
5.1. Maximum likelihood estimation
5.1.1 Random variables that are i.i.d.
5.1.2 Problems with covariates
5.2 Statistical properties of estimators
5.3 Some advanced techniques
5.3.1 EM algorithm
5.3.2 Latent variables
5.A. Finite
mixture models
6. The Bayesian Approach
6.1. Basics
6.2. Empirical Bayes
6.3. Hierarchical Bayes
6.A. General sampling algorithms
6.A.1 Metropolis algorithm
6.A.2 Some theory
6.A.3 Metropolis-Hastings algorithm
III Facing Uncertainty
7. Interval
Estimation
7.1. Uncertainty quantification
7.1.1 Bayesian version
7.1.2 Frequentist version
7.2. Main difficulty
7.3. Two useful methods
7.3.1 Likelihood ratio
7.3.2 Bootstrap
8. Tests of Significance
8.1. Basics
8.1.1 Relation to interval estimation
8.1.2
The p-value
8.2. Some challenges
8.2.1 Multiple testing
8.2.2 Six degrees of separation
8.A. Intuition of Benjamini-Hockberg
IV Appendices
A. Some Further Topics
A.1 Graphical models
A.2 Regression models
A.3 Data
collection
Epilogue
Bibliography
Index
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Mu Zhu is Professor in the Department of Statistics & Actuarial Science at the University of Waterloo, and Fellow of the American Statistical Association. He received his AB magna cum laude in applied mathematics from Harvard University, and his PhD in statistics from Stanford University. He
is currently Director of the Graduate Data Science Program at Waterloo.
