Probability Etudes in Common Lisp

Probability Etudes in Common Lisp

Mark Watson
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Table of Contents

Probability Etudes in Common Lisp

  • Preface
    • How To Read This Book
    • Running the Examples
    • Acknowledgments
  • Introduction
    • Why Study Probability?
    • What This Book Covers
    • A Note on the Code
    • How Probability Connects to the Rest of Mathematics
  • Basic Probability
    • A Brief History
    • Interpretations of Probability
    • Sample Spaces and Events
    • Set Operations on Events
    • The Event Space and the Probability Triple
    • The Kolmogorov Axioms
    • The Classical Definition of Probability
    • The Complement Rule
    • Running the Example
    • Why This Matters
    • Problem Set
  • Conditional Probability
    • Why Conditional Probability?
    • The Definition of Conditional Probability
    • The Multiplication Rule and the Chain Rule
    • Independence
    • The Law of Total Probability
    • Bayes’ Theorem
    • Simpson’s Paradox
    • Running the Example
    • Why This Matters
    • Problem Set
  • Discrete Random Variables
    • From Events to Random Variables
    • What Is a Random Variable?
    • Indicator Random Variables
    • The Probability Mass Function
    • The Cumulative Distribution Function
    • Expected Value
    • Variance and Standard Deviation
    • Standardization
    • Moments and the Moment Generating Function
    • Concentration Inequalities
    • Three Example Distributions
    • Additional Computations in the Program
    • Running the Example
    • Why This Matters
    • Problem Set
  • Binomial and Geometric Distributions
    • Bernoulli Trials
    • The Binomial Distribution
    • The Geometric Distribution
    • The Memoryless Property
    • Other Distributions from Bernoulli Trials
    • Real-World Applications
    • Running the Example
    • Why This Matters
    • Problem Set
  • Continuous Distributions
    • Why Continuous Random Variables?
    • From Probability Mass to Probability Density
    • Expectation and Variance for Continuous Variables
    • Numerical Integration
    • The Uniform Distribution
    • The Exponential Distribution
    • The Normal Distribution
    • Other Named Continuous Distributions
    • The Maximum Entropy Viewpoint
    • Transformations of Random Variables
    • Why This Matters
    • Problem Set
  • Law of Large Numbers
    • Historical Roots
    • The Sample Mean
    • The Weak Law of Large Numbers
    • The Strong Law of Large Numbers
    • Modes of Convergence
    • When the Law of Large Numbers Fails
    • Why This Matters
    • The Simulation
    • Running the Example
    • A Practical Observation
    • Problem Set
  • Central Limit Theorem
    • A Brief History
    • The Theorem
    • Why This Is Remarkable
    • Why It Is True: A Sketch via Generating Functions
    • Rate of Convergence: Berry-Esseen
    • The Simulation
    • Running the Example
    • The Theoretical Basis for Statistical Practice
    • How Large Does n Need to Be?
    • Beyond i.i.d.: Extensions of the CLT
    • Why This Matters
    • Problem Set
  • Monte Carlo Methods
    • A Short History
    • The Core Idea
    • Monte Carlo Integration
    • Estimating Pi
    • Standard Error
    • Running the Example
    • The Cost of Monte Carlo
    • Variance Reduction
    • Quasi-Random Sequences
    • Buffon’s Needle: An Older Pi Estimator
    • Applications Beyond Pi
    • Why This Matters
    • Problem Set
  • Bayesian Inference
    • Two Paradigms of Statistics
    • The Bayesian Framework
    • The Choice of Prior
    • Conjugate Priors
    • The Update Rule
    • Point Estimates and Credible Intervals
    • Prediction: The Posterior Predictive
    • The Example: Estimating a Coin’s Bias
    • Running the Example
    • The Laplace Rule of Succession
    • Prior Sensitivity
    • Why This Matters
    • Problem Set
  • Markov Chains
    • Andrey Markov and the Origin of Markov Chains
    • The Markov Property
    • A Weather Model
    • Evolving the Distribution
    • Classification of States
    • The Stationary Distribution
    • Finding the Stationary Distribution
    • Why the Chain Converges: The Spectral View
    • Running the Example
    • Hitting Times and First-Step Analysis
    • Applications of Markov Chains
    • Beyond Discrete Time
    • Why This Matters
    • Problem Set
  • An Embedded Probabilistic Programming Language
    • What Inference Computes
    • Unconstrained Space and the Jacobian
    • Three Ways to Find the Posterior
    • Forward-Mode Automatic Differentiation
    • The Distribution Library
    • Support Transforms
    • The Model DSL: defmodel, sample, observe
    • The Example Data
    • The Three Inference Engines
    • Diagnostics: ESS and R-hat
    • Running the Example
    • Wrap Up
    • Problem Set
  • Wrapping Up
    • The Arc of the Book
    • Where To Go From Here
    • The Value of Implementation
    • Final Thoughts
  • Further Reading
    • General Probability and Measure
    • Statistics, Bayesian Methods, and Machine Learning
    • Markov Chains and Monte Carlo
    • Papers Behind Chapter 11
    • Special Functions
Probability Etudes in Common Lisp/overview

Probability Etudes in Common Lisp

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Probability Etudes in Common Lisp15 chapters

Begin ›
  1. Preface

  2. Introduction

  3. Basic Probability

  4. Conditional Probability

  5. Discrete Random Variables

  6. Binomial and Geometric Distributions

  7. Continuous Distributions

  8. Law of Large Numbers

  9. Central Limit Theorem

  10. Monte Carlo Methods

  11. Bayesian Inference

  12. Markov Chains

  13. An Embedded Probabilistic Programming Language

  14. Wrapping Up

  15. Further Reading