Introduction: A Brief Tour of Concepts, Techniques, and Applications (5 weeks)

  1. The Probabilistic Method. Discrete Probability Space, Events, and the Union Bound
  2. Randomized Algorithms. Independence and Conditional Probability, Random Variables, Expectation and Conditional Expectation, Law of Total Probability, Law of Total Expectation, Derandomization Using Conditional Expectation
  3. Sampling. (Co)Variance, Moments and Deviation, Basic Concentration/Tail Inequalities (Markov, Chebyshev, Chernoff).
  4. More on Concentration Inequalities. Proofs of Markov, Chebyshev, Chernoff, Chernoff-Hoefding. The Bernstein moment generating function technique. Johnson-Lindenstrauss Lemma.

The Probabilistic Method (4 weeks)

  1. Union bound
  2. Argument from expectation, first moment method
  3. Alterations
  4. First and Second moment method. (See also Chapter 4 of the nice book by Wojciech Szpankowski.)
  5. Lovasz local lemma

Sampling, MCMC (3 weeks)

  1. Discrete Time Markov Chains
  2. #P, Approximate counting and sampling
  3. Monte Carlo and Markov Chain Monte Carlo
  4. Data streaming (Many students will present these)
  5. Compressive sampling (No time, too bad)

Your presentations

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