This chapter introduces the idea of discrete probability models and Bayesian learning. You'll express your opinion about plausible models by defining a prior probability distribution, you'll observe new information, and then, you'll update your opinion about the models by applying Bayes' theorem.

Discrete probability distributions

Most likely spin?

Chance of spinning an even number?

Constructing a spinner

Simulating spinner data

Bayes' rule

The prior

The likelihoods

The posterior

Interpreting the posterior

Sequential Bayes

Bayes with another spin

Introduction to Bayesian thinking

This chapter describes learning about a population proportion using discrete and continuous models. You'll use a beta curve to represent prior opinion about the proportion, take a sample and observe the number of successes and failures, and construct a beta posterior curve that combines both the information in the prior and in the sample. You'll then use the beta posterior curve to draw inferences about the population proportion.

Bayes with discrete models

Constructing a discrete prior

Updating by Bayes' rule

Continuous prior

Working with a beta curve

Constructing a beta prior

Updating the beta prior

Bayesian updating

Testing a claim

Bayesian inference

Interval estimates

Compare with classical methods

Bayesian probability interval

Posterior simulation

Simulating from a beta curve

Why simulate?

Learning about a binomial probability

This chapter introduces Bayesian learning about a population mean. You'll sample from a normal population with an unknown mean and a known standard deviation,  construct a normal prior to reflect your opinion about the location of the mean before sampling, and see that the posterior distribution also has a normal form with updated values of the mean and standard deviation. You'll also get more practice  drawing inferences from the posterior distribution, only this time, about a population mean.

Normal sampling model

Reaction times

Bayes with a continuous prior

A continuous prior

Comparing normal priors

Updating the normal prior

The normal posterior

Compare with classical interval

Is Jim slow?

Simulation

Predictive simulation

Prediction vs. inference

Learning about a normal mean

Suppose you're interested in comparing proportions from two populations. You take a random sample from each population and you want to learn about the difference in proportions. This chapter will illustrate the use of discrete and continuous priors to do this kind of inference. You'll use a Bayesian regression approach to learn about a mean or the difference in means when the sampling standard deviation is unknown.

Comparing two proportions

A discrete prior

Discrete posterior

Proportions with continuous priors

Independent beta priors

Posterior of proportions

Comparison of proportions

Normal model inference

Uniform prior

Learning about a percentile

Bayesian regression

Two group model

Standardized effect inference

Wrap-up and review

Bayesian comparisons

There are two schools of thought in the world of statistics, the frequentist perspective and the Bayesian perspective. At the core of the Bayesian perspective is the idea of representing your beliefs about something using the language of probability, collecting some data, then updating your beliefs based on the evidence contained in the data. This provides a convenient way of implementing the scientific method for learning about the world we live in. Bayesian statistics is increasingly popular due to recent improvements in computation, the ability to fit a wide range of models, and to produce intuitive interpretations of the results.

Introduction to R

Foundations of Probability in R

This course provides a basic introduction to Bayesian statistics in R.

Beginning Bayes in R