Probability Calculus for Data Science

The text presents basic tools of probability calculus: measurability and sigma algebras, characteristic functions and generating functions, convergence of probability distributions, the Central Limit Theorem, convergence of random variables, the Laws of Large Numbers, exponential family of distributions, multivariate normal distributions and conditional distributions. Stochastic processes included are Gaussian processes and Wiener processes (Brownian motion). The questions of data science/statistics treated by these tools are asymptotic properties of the Maximum Likelihood Estimate, Bayesian learning, bias-variance decomposition, the curse of dimensionality, EM-algorithm, logistic regression, model choice, PAC learning, supervised classification, predictive inference and probabilistic clustering. Probability Calculus for Data Science is suitable for masters’ students of data science, machine learning, statistics and financial analysis. The text is also useful for students planning to study advanced courses in signal processing and econometrics.