Bayesian probability interval

Suppose Rebekah's prior beliefs about the proportion P are represented by a beta curve with parameters 8.13 and 7.67.

After observing results from a survey, Rebekah's posterior distribution are represented by a beta curve with parameters 24.13 and 7.67. Rebekah's 90% probability interval for P is given by (0.627, 0.872).

Compare this Bayesian interval with a 90% classical interval, and pick out the false statement below!

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Beginning Bayes in R

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Since Rebekah is using prior information, the width of the Bayesian interval will be shorter than the classical interval.,The Bayesian and classical intervals will have approximately the same length.,The probability P is contained in the Bayesian interval is exactly 0.90.,The Bayesian interval using Rebekah's prior will be shorter than the Bayesian interval using a uniform prior.

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Beginning Bayes in R

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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.

Exercise 1: Discrete probability distributions Exercise 2: Most likely spin?Exercise 3: Chance of spinning an even number?Exercise 4: Constructing a spinner Exercise 5: Simulating spinner data Exercise 6: Bayes' rule Exercise 7: The prior Exercise 8: The likelihoods Exercise 9: The posterior Exercise 10: Interpreting the posterior Exercise 11: Sequential Bayes Exercise 12: Bayes with another spin

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.

Exercise 1: Bayes with discrete models Exercise 2: Constructing a discrete prior Exercise 3: Updating by Bayes' rule Exercise 4: Continuous prior Exercise 5: Working with a beta curve Exercise 6: Constructing a beta prior Exercise 7: Updating the beta prior Exercise 8: Bayesian updating Exercise 9: Testing a claim Exercise 10: Bayesian inference Exercise 11: Interval estimates Exercise 12: Compare with classical methods Exercise 13: Bayesian probability interval

Current Exercise

Exercise 14: Posterior simulation Exercise 15: Simulating from a beta curve Exercise 16: Why simulate?

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.

Exercise 1: Normal sampling model Exercise 2: Reaction times Exercise 3: Updating by Bayes' rule Exercise 4: Bayes with a continuous prior Exercise 5: A continuous prior Exercise 6: Comparing normal priors Exercise 7: Updating the normal prior Exercise 8: The normal posterior Exercise 9: Compare with classical interval Exercise 10: Is Jim slow?Exercise 11: Simulation Exercise 12: Posterior simulation Exercise 13: Predictive simulation Exercise 14: Prediction vs. inference

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.

Exercise 1: Comparing two proportions Exercise 2: A discrete prior Exercise 3: Discrete posterior Exercise 4: Proportions with continuous priors Exercise 5: Independent beta priors Exercise 6: Posterior of proportions Exercise 7: Comparison of proportions Exercise 8: Normal model inference Exercise 9: Uniform prior Exercise 10: Learning about a percentile Exercise 11: Bayesian regression Exercise 12: Two group model Exercise 13: Standardized effect inference Exercise 14: Wrap-up and review