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PART IV: Hypothesis Testing: The Heart of Statistics
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PART IV: Hypothesis Testing: The Heart of Statistics
by Will Kurt
Bayesian Statistics the Fun Way
Cover Page
Title Page
Copyright Page
Dedication
About the Author
About the Technical Reviewer
Brief Contents
Contents in Detail
Acknowledgments
Introduction
Why Learn Statistics?
What Is “Bayesian” Statistics?
What’s in This Book
Background for Reading the Book
Now Off on Your Adventure!
Part I: Introduction to Probability
1. Bayesian Thinking and Everyday Reasoning
Reasoning About Strange Experiences
Gathering More Evidence and Updating Your Beliefs
Comparing Hypotheses
Data Informs Belief; Belief Should Not Inform Data
Wrapping Up
Exercises
2. Measuring Uncertainty
What Is a Probability?
Calculating Probabilities by Counting Outcomes of Events
Calculating Probabilities as Ratios of Beliefs
Wrapping Up
Exercises
3. The Logic of Uncertainty
Combining Probabilities with AND
Combining Probabilities with OR
Wrapping Up
Exercises
4. Creating a Binomial Probability Distribution
Structure of a Binomial Distribution
Understanding and Abstracting Out the Details of Our Problem
Counting Our Outcomes with the Binomial Coefficient
Example: Gacha Games
Wrapping Up
Exercises
5. The Beta Distribution
A Strange Scenario: Getting the Data
The Beta Distribution
Reverse-Engineering the Gacha Game
Wrapping Up
Exercises
Part II: Bayesian Probability and Prior Probabilities
6. Conditional Probability
Introducing Conditional Probability
Conditional Probabilities in Reverse and Bayes’ Theorem
Introducing Bayes’ Theorem
Wrapping Up
Exercises
7. Bayes’ Theorem with LEGO
Working Out Conditional Probabilities Visually
Working Through the Math
Wrapping Up
Exercises
8. The Prior, Likelihood, and Posterior of Bayes’ Theorem
The Three Parts
Investigating the Scene of a Crime
Considering Alternative Hypotheses
Comparing Our Unnormalized Posteriors
Wrapping Up
Exercises
9. Bayesian Priors and Working with Probability Distributions
C-3PO’s Asteroid Field Doubts
Determining C-3PO’s Beliefs
Accounting for Han’s Badassery
Creating Suspense with a Posterior
Wrapping Up
Exercises
Part III: Parameter Estimation
10. Introduction to Averaging and Parameter Estimation
Estimating Snowfall
Means for Measurement vs. Means for Summary
Wrapping Up
Exercises
11. Measuring the Spread of Our Data
Dropping Coins in a Well
Finding the Mean Absolute Deviation
Finding the Variance
Finding the Standard Deviation
Wrapping Up
Exercises
12. The Normal Distribution
Measuring Fuses for Dastardly Deeds
The Normal Distribution
Solving the Fuse Problem
Some Tricks and Intuitions
“N Sigma” Events
The Beta Distribution and the Normal Distribution
Wrapping Up
Exercises
13. Tools of Parameter Estimation: The PDF, CDF, and Quantile Function
Estimating the Conversion Rate for an Email Signup List
The Probability Density Function
Introducing the Cumulative Distribution Function
The Quantile Function
Wrapping Up
Exercises
14. Parameter Estimation with Prior Probabilities
Predicting Email Conversion Rates
Taking in Wider Context with Priors
Prior as a Means of Quantifying Experience
Is There a Fair Prior to Use When We Know Nothing?
Wrapping Up
Exercises
PART IV: Hypothesis Testing: The Heart of Statistics
15. From Parameter Estimation to Hypothesis Testing: Building a Bayesian A/B Test
Setting Up a Bayesian A/B Test
Monte Carlo Simulations
Wrapping Up
Exercises
16. Introduction to the Bayes Factor and Posterior Odds: The Competition of Ideas
Revisiting Bayes’ Theorem
Building a Hypothesis Test Using the Ratio of Posteriors
Wrapping Up
Exercises
17. Bayesian Reasoning in the Twilight Zone
Bayesian Reasoning in the Twilight Zone
Using the Bayes Factor to Understand the Mystic Seer
Developing Our Own Psychic Powers
Wrapping Up
Exercises
18. When Data Doesn’t Convince You
A Psychic Friend Rolling Dice
Arguing with Relatives and Conspiracy Theorists
Wrapping Up
Exercises
19. From Hypothesis Testing to Parameter Estimation
Is the Carnival Game Really Fair?
Building a Probability Distribution
From the Bayes Factor to Parameter Estimation
Wrapping Up
Exercises
A. A Quick Introduction to R
R and RStudio
Creating an R Script
Basic Concepts in R
Functions
Random Sampling
Defining Your Own Functions
Creating Basic Plots
Exercise: Simulating a Stock Price
Summary
B. Enough Calculus to Get By
Functions
The Fundamental Theorem of Calculus
C. Answers to the Exercises
Index
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14. Parameter Estimation with Prior Probabilities
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15. From Parameter Estimation to Hypothesis Testing: Building a Bayesian A/B Test
PART IV
HYPOTHESIS TESTING: THE HEART OF STATISTICS
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