Lecture 1. Uncertainty: Control vs Exploit
1) A toy example
2) Control the uncertainty
3) Exploit the uncertainty
Lecture 2. Quantification of Uncertainty (1): Probability and Random Variables
1) Mathematical formulation of probability
2) Random variables
3) Independence
Lecture 3. Quantification of Uncertainty (2): Expectation and Variance
1) Expectation
2) Variance and standard deviation
3) Applications
Lecture 4. Universal Principle (1): Law of large numbers
1) Introduction to universality
2) Law of large numbers
3) Proof of law of large numbers
4) Applications
Lecture 5. Universal Principle (2): Central limit theorem
1) Central limit theorem
2) Applications to statistics
Lecture 6. Universal Principle (3): More on fluctuation
1) Heavy-tailed random variables
2) Large deviation principles
Lecture 7. Universal Principle (4): Random processes
1) Introduction to random processes
2) Simple random walk on a line
3) Applications to gambling
Lecture 8. Universal Principle (5): Universality of random processes
1) Universality in random walks
2) Galton-Watson tree
Lecture 9. How to use uncertainty? (1): Introduction to Markov Chains
1) Markov processes
2) Markov chains
3) Examples
Lecture 10. How to use uncertainty? (2): Universal principles of Markov chains
1) Stationary distribution
2) Universal principles for Markov chains
Lecture 11. How to use uncertainty? (3): MCMC and Cutoff phenomenon
1) Markov chain Monte Carlo (MCMC)
2) Markov chain mixing theory
3) Cutoff phenomenon
Lecture 12. How to use uncertainty? (4): Stochastic optimizations and deep learning
1) Gradient descent
2) Stochastic gradient descent
3) Mini-batch gradient descent