Statistics for Data Science

MSc Data Science WiSe 2019/20

Prof. Dr. Dirk Ostwald

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Introduction

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Data science

Computational Cognitive Neuroscience

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Initial MSc program proposal 11/2016

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Referenzen

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Data science

Data science

The art of creating meaning from data

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Data science

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Data science

Data analysis is data reduction

Raw Data Reported Data

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Data science

Data analysis is model-based

Model Formulation

Model estimation

Model Evaluation

DataReality

The Scienti-c Method

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Data science

Data analysis is an interpretative device

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Data science

Data science = Statistics = Machine learning = Artificial intelligence?

Statistics• Probabilistic models

• Theoretical analysis

• Optimality

• Asymptotics

• Science philosophy

Machine learning

• Deterministic models

• Classification

• Bayesian models

• Benchmarking

• Applications

AI (est. 2015)

• Deep learning

• Reinforcement

• Machine learning

• Data analysis

• Hype

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Statistics

Statistics

The art of creating meaning from data

and quantifying its associated uncertainty

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Statistics

Themes

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Data science

Historical development of statistics

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Statistics

Historical development of statistics

1900 Karl Pearson’s chi-square test

1908 Student’s t statistic

1925 Fisher’s Statistical Methods for Research Workers

1933 Neyman and Pearson’s optimal hypothesis testing

1937 Neyman’s confidence intervals

1950 Wald’s statistical decision theory

1950 Savage’s & de Finetti’s Bayesian decision theory

1961 Raiffa & Schlaifer’s Applied statistical decision theory

1962 Tukey’s The future of data analysis

1971 Lindley’s Bayesian statistics

1972 Cox’s proportional hazards

1979 Bootstrap and MCMC

1995 False discovery rate and LASSO

1996 Support vector machines

2000 Microarray and neuroimaging multiple testing

2010 Resurgence of neural networks as deep learning

2015 Data science

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Statistics

Course outline

Section Topic Aims

Probability (1) Probability spaces A common language

(2) Random variables A common language

(3) Joint distributions A common language

(4) Random variable transformations Core mechanism of statistics

(5) Expectation, covariance, limits Making good predictions

Frequentism (6) Foundations, maximum likelihood Creating estimators

(7) Finite-sample estimator properties How good are estimators?

(8) Asymptotic estimator properties How good are estimators?

(9) Confidence intervals How good are estimates?

(10) Hypothesis testing Making good decisions

(11) Nonparametric inference Subduing parametrics

Bayesianism (12) Foundations, conjugate inference Probabilistic inference

(13) Bayesian filtering Dynamic inference

(14) Prior distributions Objective subjectivism

(15) Markov chain Monte Carlo Approximate inference

(16) Variational inference Approximate inference

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Statistics

Some typical Machine Learning topics

• Principal and independent component analysis

• Logistic regression and linear discriminants

• Support vector machines

• Kernel methods

• Neural networks and Deep learning

• Gaussian process regression

Some typical Artificial Intelligence topics

• Reinforcement learning

• (Partially observable) Markov Decision Processes

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Statistics

Community nomenclatures

Statistic Machine Learning Meaning

Data Training data Data

Estimation Learning, Training Using data to estimate parameters

Frequentist inference - Optimal many samples methods

Bayesian inference Bayesian inference Data-based uncertainty updating

Covariates Features Structural and known data predictors

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Statistics

Central postulates of Probability theory

• Chance processes can be described mathematically.

• Mathematics can be used to make predictions about random events.

• Reasoning about uncertain events is naturally related to measuring volumes.

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Statistics

Central postulates of Frequentist inference

• Probabilities are interpreted as limiting relative frequencies and are consid-

ered objective properties of the real world.

• Parameters are fixed, unknown constants, referred to as true, but unknown

values. No probability statements are made about parameters.

• Statistical procedures are designed to have good long run frequency prop-

erties and are typically assessed by studying their sampling distributions.

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Statistics

Central postulates of Bayesian inference

• Probabilities are interpreted as degrees of belief, not limiting frequencies.

Statements like “the probability that it will rain this afternoon is 0.5” are

meaningful.

• Parameters are fixed, unknown constants, about which probabilistic state-

ments quantifying our uncertainty about their value can be made.

• Probabilistic statements about parameters are made with the help of prob-

ability distributions, from which further inferences, such as point or interval

estimates, can be derived.

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Statistics

Mathematical preliminaries

• Sets and functions

• Matrix algebra

• Calculus

• Probabilistic foundations

• Gaussian distributionsProf. Dr. Dirk Ostwald

Statistical Methods 2019/20MSc Social, Cognitive, and Affective Neuroscience

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Statistics

Key references

Applied Statistical Inference

Leonhard HeldDaniel Sabanés Bové

Likelihood and Bayes

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Statistics

Course components

Component Aims

Vorlesung Core course content, knowledge acquisition, breadth

Ubung Active participation, knowledge consolidation, depth

Study questions Focus, memorization support, examination

Exercises (Theory) Theoretical depth, self-study, oral presentations

Exercises (Programming) Intuition, Python training, application

Exam Knowledge reproduction

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Statistics

Course requirements

Vorlesung

• Written exam, pass or fail

• 30 questions, 90 min

• Study questions = exam question pool

• 17.02.2020, 06.04.2020, 20.07.20, 28.09.20

Ubung

• Attendance of 85% of the sessions

• Presentation of one theoretical or programming exercise in class

• Please sign up at bit.ly/2Vzgxx7

• Submission of Jupyter notebook of all programming exercises

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Additional course of interest WiSe 2019/20

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Additional course of interest SoSe 2020

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