# Bayesian Statistics: From Concept to Data Analysis

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### Description

About this course: This course introduces the Bayesian approach to statistics, starting with the concept of probability and moving to the analysis of data. We will learn about the philosophy of the Bayesian approach as well as how to implement it for common types of data. We will compare the Bayesian approach to the more commonly-taught Frequentist approach, and see some of the benefits of the Bayesian approach. In particular, the Bayesian approach allows for better accounting of uncertainty, results that have more intuitive and interpretable meaning, and more explicit statements of assumptions. This course combines lecture videos, computer demonstrations, readings, exercises, and discu…

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About this course: This course introduces the Bayesian approach to statistics, starting with the concept of probability and moving to the analysis of data. We will learn about the philosophy of the Bayesian approach as well as how to implement it for common types of data. We will compare the Bayesian approach to the more commonly-taught Frequentist approach, and see some of the benefits of the Bayesian approach. In particular, the Bayesian approach allows for better accounting of uncertainty, results that have more intuitive and interpretable meaning, and more explicit statements of assumptions. This course combines lecture videos, computer demonstrations, readings, exercises, and discussion boards to create an active learning experience. For computing, you have the choice of using Microsoft Excel or the open-source, freely available statistical package R, with equivalent content for both options. The lectures provide some of the basic mathematical development as well as explanations of philosophy and interpretation. Completion of this course will give you an understanding of the concepts of the Bayesian approach, understanding the key differences between Bayesian and Frequentist approaches, and the ability to do basic data analyses.

Who is this class for: This course is for people interested in learning an alternative to the Frequentist approach that is typically taught in statistics classes. The course covers both concepts and basic computing, and so it is applicable both to people doing data analysis as well as people who read the analysis of others, such as decision makers. This course expects that learners have previous exposure to statistics at the introductory level or higher, and previous exposure to calculus. In both cases, the expectation is that concepts have been seen previously, possibly many years earlier, but that the details may have been forgotten.

Created by:  University of California, Santa Cruz
• Taught by:  Herbert Lee, Professor

Applied Mathematics and Statistics
Level Intermediate Commitment Four weeks of study, two-five hours/week depending on your familiarity with mathematical statistics. Language English How To Pass Pass all graded assignments to complete the course. User Ratings 4.5 stars Average User Rating 4.5See what learners said Coursework

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Syllabus

WEEK 1

Probability and Bayes' Theorem

In this module, we review the basics of probability and Bayes’ theorem. In Lesson 1, we introduce the different paradigms or definitions of probability and discuss why probability provides a coherent framework for dealing with uncertainty. In Lesson 2, we review the rules of conditional probability and introduce Bayes’ theorem. Lesson 3 reviews common probability distributions for discrete and continuous random variables.

8 videos, 4 readings expand

1. Video: Course introduction
2. Reading: Module 1 objectives, assignments, and supplementary materials
3. Reading: Background for Lesson 1
4. Video: Lesson 1.1 Classical and frequentist probability
5. Video: Lesson 1.2 Bayesian probability and coherence
6. Discussion Prompt: Objectivity
7. Video: Lesson 2.1 Conditional probability
8. Video: Lesson 2.2 Bayes' theorem
9. Reading: Supplementary material for Lesson 2
10. Video: Lesson 3.1 Bernoulli and binomial distributions
11. Video: Lesson 3.2 Uniform distribution
12. Video: Lesson 3.3 Exponential and normal distributions
13. Reading: Supplementary material for Lesson 3

Graded: Lesson 1
Graded: Lesson 2
Graded: Lesson 3.1
Graded: Lesson 3.2-3.3
Graded: Module 1 Honors

WEEK 2

Statistical Inference

This module introduces concepts of statistical inference from both frequentist and Bayesian perspectives. Lesson 4 takes the frequentist view, demonstrating maximum likelihood estimation and confidence intervals for binomial data. Lesson 5 introduces the fundamentals of Bayesian inference. Beginning with a binomial likelihood and prior probabilities for simple hypotheses, you will learn how to use Bayes’ theorem to update the prior with data to obtain posterior probabilities. This framework is extended with the continuous version of Bayes theorem to estimate continuous model parameters, and calculate posterior probabilities and credible intervals.

11 videos, 5 readings expand

1. Reading: Module 2 objectives, assignments, and supplementary materials
2. Reading: Background for Lesson 4
3. Video: Lesson 4.1 Confidence intervals
4. Video: Lesson 4.2 Likelihood function and maximum likelihood
5. Video: Lesson 4.3 Computing the MLE
6. Video: Lesson 4.4 Computing the MLE: examples
7. Reading: Supplementary material for Lesson 4
8. Video: Introduction to R
9. Video: Plotting the likelihood in R
10. Video: Plotting the likelihood in Excel
11. Reading: Background for Lesson 5
12. Video: Lesson 5.1 Inference example: frequentist
13. Video: Lesson 5.2 Inference example: Bayesian
14. Video: Lesson 5.3 Continuous version of Bayes' theorem
15. Video: Lesson 5.4 Posterior intervals
16. Discussion Prompt: Confidence intervals and credible intervals
17. Reading: Supplementary material for Lesson 5

Graded: Lesson 4
Graded: Lesson 5.1-5.2
Graded: Lesson 5.3-5.4
Graded: Module 2 Honors

WEEK 3

Priors and Models for Discrete Data

In this module, you will learn methods for selecting prior distributions and building models for discrete data. Lesson 6 introduces prior selection and predictive distributions as a means of evaluating priors. Lesson 7 demonstrates Bayesian analysis of Bernoulli data and introduces the computationally convenient concept of conjugate priors. Lesson 8 builds a conjugate model for Poisson data and discusses strategies for selection of prior hyperparameters.

9 videos, 2 readings expand

1. Reading: Module 3 objectives, assignments, and supplementary materials
2. Video: Lesson 6.1 Priors and prior predictive distributions
3. Video: Lesson 6.2 Prior predictive: binomial example
4. Video: Lesson 6.3 Posterior predictive distribution
5. Video: Lesson 7.1 Bernoulli/binomial likelihood with uniform prior
6. Video: Lesson 7.2 Conjugate priors
7. Video: Lesson 7.3 Posterior mean and effective sample size
8. Video: Data analysis example in R
9. Video: Data analysis example in Excel
10. Discussion Prompt: Prior elicitation
11. Reading: R and Excel code from example analysis
12. Video: Lesson 8.1 Poisson data

Graded: Lesson 6
Graded: Lesson 7
Graded: Lesson 8
Graded: Module 3 Honors

WEEK 4

Models for Continuous Data

This module covers conjugate and objective Bayesian analysis for continuous data. Lesson 9 presents the conjugate model for exponentially distributed data. Lesson 10 discusses models for normally distributed data, which play a central role in statistics. In Lesson 11, we return to prior selection and discuss ‘objective’ or ‘non-informative’ priors. Lesson 12 presents Bayesian linear regression with non-informative priors, which yield results comparable to those of classical regression.

9 videos, 5 readings expand

1. Reading: Module 4 objectives, assignments, and supplementary materials
2. Video: Lesson 9.1 Exponential data
3. Video: Lesson 10.1 Normal likelihood with variance known
4. Video: Lesson 10.2 Normal likelihood with variance unknown
5. Reading: Supplementary material for Lesson 10
6. Video: Lesson 11.1 Non-informative priors
7. Video: Lesson 11.2 Jeffreys prior
8. Discussion Prompt: A non-informative prior
9. Reading: Supplementary material for Lesson 11
10. Reading: Background for Lesson 12
11. Video: Linear regression in R
12. Video: Linear regression in Excel (Analysis ToolPak)
13. Video: Linear regression in Excel (StatPlus by AnalystSoft)
14. Reading: R and Excel code for regression
15. Video: Conclusion

Graded: Lesson 9
Graded: Lesson 10
Graded: Lesson 11
Graded: Regression
Graded: Module 4 Honors
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