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Python for Probability, Statistics, and Machine Learning.pdf
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This book covers the key ideas that link probability, statistics, and machine learning illustrated using Python modules in these areas. The entire text, including all the figures and numerical results, is reproducible using the Python codes and their associated Jupyter/IPython notebooks, which are provided as supplementary downloads. The author develops key intuitions in machine learning by working meaningful examples using multiple analytical methods and Python codes, thereby connecting theoretical concepts to concrete implementations. Modern Python modules like Pandas, Sympy, and Scikit-learn are applied to simulate and visualize important machine learning concepts like the bias/variance trade-off, cross-validation, and regularization. Many abstract mathematical ideas, such as convergence in probability theory, are developed and illustrated with numerical examples. This book is suitable for anyone with an undergraduate-level exposure to probability, statistics, or machine learning and with rudimentary knowledge of Python programming.
Contents
1 Getting Started with Scientific Python
1.1 Installation and Setup
1.2 Numpy
1.2.1 Numpy Arrays and Memory
1.2.2 Numpy Matrices
1.2.3 Numpy Broadcasting
1.2.4 Numpy Masked Arrays
1.2.5 Numpy Optimizations and Prospectus
1.3 Matplotlib
1.3.1 Alternatives to Matplotlib
1.3.2 Extensions to Matplotlib
1.4 IPython
1.4.1 IPython Notebook
1.5 Scipy
1.6 Pandas
1.6.1 Series
1.6.2 Dataframe
1.7 Sympy
1.8 Interfacing with Compiled Libraries
1.9 Integrated Development Environments
1.10 Quick Guide to Performance and Parallel Programming
1.11 Other Resources
References
2 Probability
2.1 Introduction
2.1.1 Understanding Probability Density
2.1.2 Random Variables
2.1.3 Continuous Random Variables
2.1.4 Transformation of Variables Beyond Calculus
2.1.5 Independent Random Variables
2.1.6 Classic Broken Rod Example
2.2 Projection Methods
2.2.1 Weighted Distance
2.3 Conditional Expectation as Projection
2.3.1 Appendix
2.4 Conditional Expectation and Mean Squared Error
2.5 Worked Examples of Conditional Expectation and Mean Square
Error Optimization
2.5.1 Example
2.5.2 Example
2.5.3 Example
2.5.4 Example
2.5.5 Example
2.5.6 Example
2.6 Information Entropy
2.6.1 Information Theory Concepts
2.6.2 Properties of Information Entropy
2.6.3 Kullback-Leibler Divergence
2.7 Moment Generating Functions
2.8 Monte Carlo Sampling Methods
2.8.1 Inverse CDF Method for Discrete Variables
2.8.2 Inverse CDF Method for Continuous Variables
2.8.3 Rejection Method
2.9 Useful Inequalities
2.9.1 Markov’s Inequality
2.9.2 Chebyshev’s Inequality
2.9.3 Hoeffding’s Inequality
References
3 Statistics
3.1 Introduction
3.2 Python Modules for Statistics
3.2.1 Scipy Statistics Module
3.2.2 Sympy Statistics Module
3.2.3 Other Python Modules for Statistics
3.3 Types of Convergence
3.3.1 Almost Sure Convergence
3.3.2 Convergence in Probability
3.3.3 Convergence in Distribution
3.3.4 Limit Theorems
3.4 Estimation Using Maximum Likelihood
3.4.1 Setting Up the Coin Flipping Experiment
3.4.2 Delta Method
3.5 Hypothesis Testing and P-Values
3.5.1 Back to the Coin Flipping Example
3.5.2 Receiver Operating Characteristic
3.5.3 P-Values
3.5.4 Test Statistics
3.5.5 Testing Multiple Hypotheses
3.6 Confidence Intervals
3.7 Linear Regression
3.7.1 Extensions to Multiple Covariates
3.8 Maximum A-Posteriori
3.9 Robust Statistics
3.10 Bootstrapping
3.10.1 Parametric Bootstrap
3.11 Gauss Markov
3.12 Nonparametric Methods
3.12.1 Kernel Density Estimation
3.12.2 Kernel Smoothing
3.12.3 Nonparametric Regression Estimators
3.12.4 Nearest Neighbors Regression
3.12.5 Kernel Regression
3.12.6 Curse of Dimensionality
References
4 Machine Learning
4.1 Introduction
4.2 Python Machine Learning Modules
4.3 Theory of Learning
4.3.1 Introduction to Theory of Machine Learning
4.3.2 Theory of Generalization
4.3.3 Worked Example for Generalization/Approximation
Complexity
4.3.4 Cross-Validation
4.3.5 Bias and Variance
4.3.6 Learning Noise
4.4 Decision Trees
4.4.1 Random Forests
4.5 Logistic Regression
4.5.1 Generalized Linear Models
4.6 Regularization
4.6.1 Ridge Regression
4.6.2 Lasso
4.7 Support Vector Machines
4.7.1 Kernel Tricks
4.8 Dimensionality Reduction
4.8.1 Independent Component Analysis
4.9 Clustering
4.10 Ensemble Methods
4.10.1 Bagging
4.10.2 Boosting
References
Index
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