Posterior credible intervals

Let's focus on slope parameter \(b\), the rate of change in weight over height. The posterior mean of \(b\) reflects the trend in the posterior model of the slope. In contrast, a posterior credible interval provides a range of posterior plausible slope values, thus reflects posterior uncertainty about \(b\). For example, the 95% credible interval for \(b\) ranges from the 2.5th to the 97.5th quantile of the \(b\) posterior. Thus there's a 95% (posterior) chance that \(b\) is in this range.

You will use RJAGS simulation output to approximate credible intervals for \(b\). The 100,000 iteration RJAGS simulation of the posterior, weight_sim_big, is in your workspace along with a data frame of the Markov chain output, weight_chains.

This is a part of the course

“Bayesian Modeling with RJAGS”

Exercise instructions

Obtain summary() statistics of the weight_sim_big chains.
The 2.5% and 97.5% posterior quantiles for \(b\) are reported in Table 2 of the summary(). Apply quantile() to the raw weight_chains to verify these calculations. Save this as ci_95 and print it.
Similarly, use the weight_chains data to construct a 90% credible interval for \(b\). Save this as ci_90 and print it.
Construct a density plot of the \(b\) Markov chain values. Superimpose vertical lines representing the 90% credible interval for \(b\) using geom_vline() with xintercept = ci_90.

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Summarize the posterior Markov chains


# Calculate the 95% posterior credible interval for b
ci_95 <- quantile(___, probs = c(___, ___))
ci_95

# Calculate the 90% posterior credible interval for b
ci_90 <- ___
ci_90

# Mark the 90% credible interval 
ggplot(___, aes(x = ___)) + 
    geom_density() + 
    geom_vline(xintercept = ___, color = "red")

Edit and Run Code

This exercise is part of the course

Bayesian Modeling with RJAGS

AdvancedSkill Level

4.7+

In this course, you'll learn how to implement more advanced Bayesian models using RJAGS.

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Bayesian models combine prior insights with insights from observed data to form updated, posterior insights about a parameter. In this chapter, you will review these Bayesian concepts in the context of the foundational Beta-Binomial model for a proportion parameter. You will also learn how to use the rjags package to define, compile, and simulate this model in R.

Exercise 1: The prior model Exercise 2: Simulating a Beta prior Exercise 3: Comparing & contrasting Beta priors Exercise 4: Which prior?Exercise 5: Data & the likelihood Exercise 6: Simulating the dependence of X on p Exercise 7: Approximating the likelihood function Exercise 8: Interpreting the likelihood function Exercise 9: The posterior model Exercise 10: Define, compile, and simulate Exercise 11: Updating the posterior Exercise 12: Influence of the prior & data on the posterior

The two-parameter Normal-Normal Bayesian model provides a simple foundation for Normal regression models. In this chapter, you will engineer the Normal-Normal and define, compile, and simulate this model using rjags. You will also explore the magic of the Markov chain mechanics behind rjags simulation.

Exercise 1: The Normal-Normal model Exercise 2: Normal-Normal priors Exercise 3: Sleep study data Exercise 4: Insights from the prior and data Exercise 5: Simulating the Normal-Normal in RJAGS Exercise 6: Define, compile, & simulate the Normal-Normal Exercise 7: Posterior insights on sleep deprivation Exercise 8: Markov chains Exercise 9: Storing Markov chains Exercise 10: Markov chain trace plots Exercise 11: Markov chain density plots Exercise 12: Markov chain diagnostics & reproducibility Exercise 13: Multiple chains Exercise 14: Naive standard errors Exercise 15: Reproducibility

In this chapter, you will extend the Normal-Normal model to a simple Bayesian regression model. Within this context, you will explore how to use rjags simulation output to conduct posterior inference. Specifically, you will construct posterior estimates of regression parameters using posterior means & credible intervals, you will test hypotheses using posterior probabilities, and you will construct posterior predictive distributions for new observations.

Exercise 1: A simple Bayesian regression model Exercise 2: Regression priors Exercise 3: Visualizing the regression priors Exercise 4: Weight & height data Exercise 5: Bayesian regression in RJAGS Exercise 6: Define, compile, & simulate the regression model Exercise 7: Regression Markov chains Exercise 8: Posterior estimation & inference Exercise 9: Posterior point estimates Exercise 10: Posterior credible intervals

Exercise 11: Posterior probabilities Exercise 12: Posterior prediction Exercise 13: Inference for the posterior trend Exercise 14: Calculating posterior predictions Exercise 15: Posterior predictive distribution

In this final chapter, you will generalize the simple Normal regression model for application in broader contexts. You will incorporate categorical predictors, engineer a multivariate regression model with two predictors, and finally extend this methodology to Poisson multivariate regression models for count variables.

Exercise 1: Bayesian regression with a categorical predictor Exercise 2: RailTrail sample data Exercise 3: RJAGS simulation with categorical variables Exercise 4: Interpreting categorical coefficients Exercise 5: Inference for volume by weekday Exercise 6: Multivariate Bayesian regression Exercise 7: Re-examining the RailTrail data Exercise 8: RJAGS simulation for multivariate regression Exercise 9: Interpreting multivariate regression parameters Exercise 10: Posterior inference for multivariate regression Exercise 11: Bayesian Poisson regression Exercise 12: RJAGS simulation for Poisson regression Exercise 13: Plotting the Poisson regression model Exercise 14: Inference for the Poisson rate parameter Exercise 15: Poisson posterior prediction Exercise 16: Conclusion