Probability and statistics

See also the Probability Puzzles section of the blog, and the other probability courses.

General probability
  1. Random variables as vectors
  2. Covariance matrix and Mahalanobis distance
  3. Moments as tensors; tensor notation for moments
  4. Moment generating functions, aka Why/when moments suffice
  5. Probabilistic convergence
  6. Probabilistic inequalities
Statistics
Statistics can be understood as the study of the applications of Bayes's theorem with different priors.
  1. Introduction to Bayesian inference
  2. Uniform prior: Maximum Likelihood Estimation
  3. Invariant prior: Confidence Regions and Hypothesis tests
  4. I don't believe p-hacking is a problem. + "at least one of them is a boy" (analogous to "at least 10 heads" without discussing the number of trials)
  5. The meaningless bureaucracy of frequentism
  6. Probability and surprise 
  7. Sufficient statistics
a. Frequentist definition of probability, b. Frequentist claim that you can't actually talk about probabilities (based on philosophical claim that the parameter is fixed) ... (a) Only way to formalize that is with probability theory (b) Ultimately you are always talking about probability. 

Sample statistics
  1. Sample statistics: Central limit theorem
  2. Sample statistics: Order statistics
  3. Sample statistics: Extreme value theory
Estimators
  1. Bayes's theorem, estimators and a theory of Amazon ratings
  2. Is overfitting about estimator variance? 
  3. Fisher info, Bernstein-von Mises, Rao-Cramer
  4. Degrees of freedom [1] [2]
  5. Robust statistics

Reference: special distributions
This is not to be read "in order" or after the above section or whatever -- it's just some reference material on special cases.
  1. Normal (inflection pt, rot. invar., etc.)
  2. Bernoulli and Poisson processes
  3. The Dirichlet (also Beta) distribution
  4. Distribution of means with non-vanishing variance: Cauchy
  5. Heavy tails, perhaps some generalisations

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