Overview. DS GA 1002 Probability and Statistics for Data Science. Carlos Fernandez-Granda
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1 Overview DS GA 1002 Probability and Statistics for Data Science Carlos Fernandez-Granda
2 Probability and statistics Probability: Framework for dealing with uncertainty Statistics: Framework for extracting information from data making probabilistic assumptions
3 Probability Probability basics: Probability spaces, conditional probability, independence Random variables: continuous/discrete, important distributions, generating random variables (rejection sampling) Multivariate random variables: random vectors, continuous/discrete, independence (conditional independence, graphical models), generating multivariate random variables
4 Probability Expectation: expectation operator, mean, variance, Markov and Chebyshev inequalities, covariance, covariance matrices, conditional expectation Random processes: Definition, mean, autocovariance, important processes (iid sequences, Gaussian, Poisson, random walk) Convergence of random sequences: Types of convergence (in probability/distribution), law of large numbers, central limit theorem, Monte Carlo simulation Markov chains: Definition, recurrence, periodicity, convergence, Markov chain Monte Carlo (Metropolis-Hastings)
5 Statistics Descriptive statistics: Histogram, empirical mean/variance, order statistics, empirical covariance, empirical covariance matrix (principal component analysis) Frequentist statistics: iid sampling, mean square error, consistency, nonparametric model estimation (kernel density estimation), parametric model estimation (method of moments, maximum likelihood)
6 Statistics Bayesian statistics: Bayesian parametric models, conjugate priors, Bayesian estimators (minimum MSE estimator, maximum a posteriori) Hypothesis testing: Hypothesis-testing framework, parametric testing, nonparametric testing (permutation test), multiple testing Linear regression: Linear models, least-squares estimation, overfitting
7 Why should I take this course?
8 To understand probabilistic models
9 United States presidential election Indirect election, citizens of the US cast ballots for electors in the Electoral College These electors vote for the President and Vice President Number of electors per state = members of Congress (Washington D.C. gets 3) Except in Maine and Nebraska, all electors in a state go to the candidate who wins the state
10 538 probabilistic model (from fivethirtyeight.com) Aim: Predict the election result using poll data Probabilistic models allow to take into account that Polls have different sample sizes Some pollsters are unreliable In some states there may be few polls (especially at the start of the campaign) Historic trends in each state are important Polls from states with similar demographics are correlated Additional information (approval ratings, contributions, party identification,... ) can be useful In addition, probabilistic models quantify the uncertainty of the prediction
11 538 probabilistic model (from fivethirtyeight.com)
12 To understand statistical methodology
13 Polio vaccine Poliomyelitis is an infectious disease, which induces paralysis and can be lethal It has almost been eradicated by vaccination (98 cases in 2015 from in 1988) The first vaccine was developed in 1952 by Jonas Salk and collaborators Two experiments were carried out to evaluate whether the vaccine was effective
14 Polio vaccine Experiment 1: Students in 2nd grade with consent of their parents were vaccinated. Students in 1st and 3rd grade were not. Experiment 2: A group of children, whose parents consented, was randomly divided in half to form the treatment and control groups. Experiment 1 Experiment 2 Size Rate Treatment Control No consent Size Rate Treatment Control No consent
15 To understand machine-learning algorithms
16 Quadratic discriminant analysis Labeled data
17 Quadratic discriminant analysis Aim: Classify unlabeled examples
18 Quadratic discriminant analysis Quadratic discriminant analysis fits a Gaussian distribution to each class
19 Quadratic discriminant analysis Results: red (99.9 %), blue (55.8 %), blue (97.2 %)
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