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<!DOCTYPE html>
<html lang="en">
<head>
<title>Bayesian Statistics Exercises in R: 20 Practice Problems</title>
<meta charset="utf-8">
<meta name="Description" content="Twenty Bayesian statistics R exercises: conjugate priors, posteriors, rstanarm, brms, MCMC diagnostics, LOO model comparison. Hidden solutions.">
<meta name="Keywords" content="Bayesian R exercises, brms exercises, rstanarm R, Bayesian regression R, MCMC R exercises, posterior R">
<meta name="Distribution" content="Global">
<meta name="Author" content="Selva Prabhakaran">
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<div id="sidebar-nav"><div class="continue-chip" data-continue-chip><span class="chip-label">Continue reading</span><a href="#" data-continue-link></a></div><div class="sidebar-tabs" role="tablist"><button class="sidebar-tab active" data-tab="posts" type="button" role="tab" onclick="var n=this.dataset.tab;document.querySelectorAll('.sidebar-tab').forEach(function(x){x.classList.toggle('active',x.dataset.tab===n)});document.querySelectorAll('.sidebar-panel').forEach(function(p){p.classList.toggle('active',p.dataset.panel===n)});try{localStorage.setItem('rstat_sidebar_tab',n)}catch(e){}">Posts</button><button class="sidebar-tab" data-tab="tools" type="button" role="tab" onclick="var n=this.dataset.tab;document.querySelectorAll('.sidebar-tab').forEach(function(x){x.classList.toggle('active',x.dataset.tab===n)});document.querySelectorAll('.sidebar-panel').forEach(function(p){p.classList.toggle('active',p.dataset.panel===n)});try{localStorage.setItem('rstat_sidebar_tab',n)}catch(e){}">Tools</button></div><div class="sidebar-panel active" data-panel="posts"><ul class="sidebar-menu list-unstyled"><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Learn R<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Getting Started</li><li data-subkey="sec0sub1"><a href="/Is-R-Worth-Learning-in-2026.html"><span class="progress-dot"></span>Is R Worth Learning?</a></li><li data-subkey="sec0sub1"><a href="/Install-R-and-RStudio-2026.html"><span class="progress-dot"></span>Install R & RStudio</a></li><li data-subkey="sec0sub1"><a href="/RStudio-IDE-Tour.html"><span class="progress-dot"></span>RStudio IDE Tour</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> R Fundamentals</li><li data-subkey="sec0sub2"><a href="/R-Syntax-101.html"><span class="progress-dot"></span>R Syntax 101</a></li><li data-subkey="sec0sub2"><a href="/R-Data-Types.html"><span class="progress-dot"></span>R Data Types</a></li><li data-subkey="sec0sub2"><a href="/R-Vectors.html"><span class="progress-dot"></span>R Vectors</a></li><li data-subkey="sec0sub2"><a href="/R-Matrices.html"><span class="progress-dot"></span>R Matrices</a></li><li data-subkey="sec0sub2"><a href="/R-Factors.html"><span class="progress-dot"></span>R Factors</a></li><li data-subkey="sec0sub2"><a href="/R-Data-Frames.html"><span class="progress-dot"></span>R Data Frames</a></li><li data-subkey="sec0sub2"><a href="/R-Lists.html"><span class="progress-dot"></span>R Lists</a></li><li data-subkey="sec0sub2"><a href="/R-Control-Flow.html"><span class="progress-dot"></span>R Control Flow</a></li><li data-subkey="sec0sub2"><a href="/R-Special-Values.html"><span class="progress-dot"></span>R Special Values</a></li><li data-subkey="sec0sub2"><a href="/R-Type-Coercion.html"><span class="progress-dot"></span>R Type Coercion</a></li><li data-subkey="sec0sub2"><a href="/R-Functions.html"><span class="progress-dot"></span>Writing R Functions</a></li><li data-subkey="sec0sub2"><a href="/R-Beginner-Exercises-quiz.html"><span class="progress-dot"></span>R Fundamentals Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Working Effectively</li><li data-subkey="sec0sub3"><a href="/R-Subsetting.html"><span class="progress-dot"></span>R Subsetting</a></li><li data-subkey="sec0sub3"><a href="/Getting-Help-in-R.html"><span class="progress-dot"></span>Getting Help in R</a></li><li data-subkey="sec0sub3"><a href="/R-Project-Structure.html"><span class="progress-dot"></span>R Project Structure</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> R Career & Resources</li><li data-subkey="sec0sub4"><a href="/R-vs-Python.html"><span class="progress-dot"></span>R vs Python</a></li><li data-subkey="sec0sub4"><a href="/How-to-Learn-R.html"><span class="progress-dot"></span>How to Learn R</a></li><li data-subkey="sec0sub4"><a href="/R-for-Excel-Users.html"><span class="progress-dot"></span>R for Excel Users</a></li><li data-subkey="sec0sub4"><a href="/R-Interview-Questions.html"><span class="progress-dot"></span>R Interview Questions</a></li><li data-subkey="sec0sub4"><a href="/R-Interview-Questions-quiz.html"><span class="progress-dot"></span>R Interview Readiness Quiz</a></li><li data-subkey="sec0sub4"><a href="/R-Cheat-Sheet.html"><span class="progress-dot"></span>R Cheat Sheet</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Professional R</li><li data-subkey="sec0sub5"><a href="/Data-Ethics-in-R.html"><span class="progress-dot"></span>Data Ethics</a></li><li data-subkey="sec0sub5"><a href="/Bias-in-Data-and-Models.html"><span class="progress-dot"></span>Bias in Data & Models</a></li><li data-subkey="sec0sub5"><a href="/Reproducibility-Crisis.html"><span class="progress-dot"></span>Reproducibility</a></li><li data-subkey="sec0sub5"><a href="/Data-Privacy-in-R.html"><span class="progress-dot"></span>Data Privacy</a></li><li data-subkey="sec0sub5"><a href="/Communicating-Uncertainty.html"><span class="progress-dot"></span>Communicating Uncertainty</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Data Wrangling<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Import & Setup</li><li data-subkey="sec1sub1"><a href="/Importing-Data-in-R.html"><span class="progress-dot"></span>Importing Data</a></li><li data-subkey="sec1sub1"><a href="/R-Pipe-Operator.html"><span class="progress-dot"></span>Pipe Operator</a></li><li data-subkey="sec1sub1"><a href="/Tidy-Data-in-R.html"><span class="progress-dot"></span>Tidy Data</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> dplyr Essentials</li><li data-subkey="sec1sub2"><a href="/dplyr-filter-select.html"><span class="progress-dot"></span>dplyr filter & select</a></li><li data-subkey="sec1sub2"><a href="/dplyr-mutate-rename.html"><span class="progress-dot"></span>dplyr mutate & rename</a></li><li data-subkey="sec1sub2"><a href="/dplyr-group-by-summarise.html"><span class="progress-dot"></span>dplyr group_by & summarise</a></li><li data-subkey="sec1sub2"><a href="/dplyr-arrange-slice.html"><span class="progress-dot"></span>dplyr arrange & slice</a></li><li data-subkey="sec1sub2"><a href="/dplyr-across.html"><span class="progress-dot"></span>dplyr across()</a></li><li data-subkey="sec1sub2"><a href="/dplyr-case-when.html"><span class="progress-dot"></span>dplyr case_when()</a></li><li data-subkey="sec1sub2"><a href="/dplyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>dplyr Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Join & Reshape</li><li data-subkey="sec1sub3"><a href="/R-Joins.html"><span class="progress-dot"></span>R Joins</a></li><li data-subkey="sec1sub3"><a href="/pivot_longer-pivot_wider-Reshape-Data-in-R.html"><span class="progress-dot"></span>pivot_longer & pivot_wider</a></li><li data-subkey="sec1sub3"><a href="/tidyr-separate-unite-Split-Combine-Columns-in-R.html"><span class="progress-dot"></span>separate() & unite()</a></li><li data-subkey="sec1sub3"><a href="/tidyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>tidyr Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Clean & Quality</li><li data-subkey="sec1sub4"><a href="/Missing-Values-in-R-Detect-Count-Remove-Impute-NA.html"><span class="progress-dot"></span>Missing Values (NA)</a></li><li data-subkey="sec1sub4"><a href="/Data-Quality-Checking-in-R.html"><span class="progress-dot"></span>Data Quality Checking</a></li><li data-subkey="sec1sub4"><a href="/janitor-Package-in-R.html"><span class="progress-dot"></span>janitor Package</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Strings & Dates</li><li data-subkey="sec1sub5"><a href="/stringr-in-R.html"><span class="progress-dot"></span>stringr</a></li><li data-subkey="sec1sub5"><a href="/R-Regex-stringr-Pattern-Matching.html"><span class="progress-dot"></span>Regex Patterns</a></li><li data-subkey="sec1sub5"><a href="/lubridate-in-R.html"><span class="progress-dot"></span>lubridate</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Scale & Connect</li><li data-subkey="sec1sub6"><a href="/DBI-in-R.html"><span class="progress-dot"></span>DBI & Databases</a></li><li data-subkey="sec1sub6"><a href="/DuckDB-in-R.html"><span class="progress-dot"></span>DuckDB & duckplyr</a></li><li data-subkey="sec1sub6"><a href="/Web-Scraping-in-R-with-rvest.html"><span class="progress-dot"></span>Web Scraping (rvest)</a></li><li data-subkey="sec1sub6"><a href="/REST-APIs-in-R-with-httr2.html"><span class="progress-dot"></span>REST APIs (httr2)</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Visualization<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> ggplot2 Foundations</li><li data-subkey="sec2sub1"><a href="/ggplot2-Grammar-of-Graphics.html"><span class="progress-dot"></span>Grammar of Graphics</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Getting-Started.html"><span class="progress-dot"></span>ggplot2 Getting Started</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Aesthetics-aes-Map-Data.html"><span class="progress-dot"></span>ggplot2 Aesthetics (aes)</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Colours.html"><span class="progress-dot"></span>ggplot2 Colours</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Scales.html"><span class="progress-dot"></span>ggplot2 Scales</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Themes-in-R.html"><span class="progress-dot"></span>ggplot2 Themes</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Labels-and-Annotations.html"><span class="progress-dot"></span>Labels & Annotations</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Facets.html"><span class="progress-dot"></span>ggplot2 Facets</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Exercises-in-R-quiz.html"><span class="progress-dot"></span>ggplot2 Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Core Charts</li><li data-subkey="sec2sub2"><a href="/ggplot2-Scatter-Plots.html"><span class="progress-dot"></span>Scatter Plots</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Line-Charts.html"><span class="progress-dot"></span>Line Charts</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Bar-Charts.html"><span class="progress-dot"></span>Bar Charts</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Distribution-Charts.html"><span class="progress-dot"></span>Distribution Charts</a></li><li data-subkey="sec2sub2"><a href="/Error-Bars-in-R.html"><span class="progress-dot"></span>Error Bars</a></li><li data-subkey="sec2sub2"><a href="/geom_smooth-in-R.html"><span class="progress-dot"></span>geom_smooth()</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Distributions & Groups</li><li data-subkey="sec2sub3"><a href="/Violin-Plot-in-R.html"><span class="progress-dot"></span>Violin Plot</a></li><li data-subkey="sec2sub3"><a href="/Ridgeline-Plot-in-R.html"><span class="progress-dot"></span>Ridgeline Plot</a></li><li data-subkey="sec2sub3"><a href="/Lollipop-Chart-in-R.html"><span class="progress-dot"></span>Lollipop Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Relationships</li><li data-subkey="sec2sub4"><a href="/Bubble-Chart-in-R.html"><span class="progress-dot"></span>Bubble Chart</a></li><li data-subkey="sec2sub4"><a href="/Heatmap-in-R.html"><span class="progress-dot"></span>Heatmap in R</a></li><li data-subkey="sec2sub4"><a href="/Correlation-Matrix-Plot-in-R.html"><span class="progress-dot"></span>Correlation Matrix</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Advanced Charts</li><li data-subkey="sec2sub5"><a href="/Pie-Donut-Chart-in-R.html"><span class="progress-dot"></span>Pie & Donut Chart</a></li><li data-subkey="sec2sub5"><a href="/Treemap-in-R.html"><span class="progress-dot"></span>Treemap</a></li><li data-subkey="sec2sub5"><a href="/Waffle-Chart-in-R.html"><span class="progress-dot"></span>Waffle Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Exploratory Analysis</li><li data-subkey="sec2sub6"><a href="/Exploratory-Data-Analysis-in-R.html"><span class="progress-dot"></span>EDA (7-Step Framework)</a></li><li data-subkey="sec2sub6"><a href="/Univariate-EDA-in-R.html"><span class="progress-dot"></span>Univariate EDA</a></li><li data-subkey="sec2sub6"><a href="/Bivariate-EDA-in-R.html"><span class="progress-dot"></span>Bivariate EDA</a></li><li data-subkey="sec2sub6"><a href="/Descriptive-Statistics-in-R.html"><span class="progress-dot"></span>Descriptive Statistics</a></li><li data-subkey="sec2sub6"><a href="/Correlation-Analysis-in-R.html"><span class="progress-dot"></span>Correlation Analysis</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub7" data-collapsed="false"><span class="subsec-chevron">▼</span> Interactive & Maps</li><li data-subkey="sec2sub7"><a href="/Combining-ggplot2-with-plotly.html"><span class="progress-dot"></span>ggplot2 + plotly Interactive</a></li><li data-subkey="sec2sub7"><a href="/Interactive-Maps-in-R-with-leaflet.html"><span class="progress-dot"></span>Leaflet Interactive Maps</a></li><li data-subkey="sec2sub7"><a href="/Spatial-Data-in-R-with-sf.html"><span class="progress-dot"></span>Spatial Data (sf)</a></li><li data-subkey="sec2sub7"><a href="/Choropleth-Maps-in-R.html"><span class="progress-dot"></span>Choropleth Maps (sf)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub8" data-collapsed="false"><span class="subsec-chevron">▼</span> Customization & Reference</li><li data-subkey="sec2sub8"><a href="/ggplot2-Legends-in-R.html"><span class="progress-dot"></span>ggplot2 Legends</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-Secondary-Axis.html"><span class="progress-dot"></span>Secondary Axis</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-Log-Scale.html"><span class="progress-dot"></span>Log Scale</a></li><li data-subkey="sec2sub8"><a href="/patchwork-Package.html"><span class="progress-dot"></span>patchwork (Combine Plots)</a></li><li data-subkey="sec2sub8"><a href="/Publication-Quality-Figures-in-R.html"><span class="progress-dot"></span>Publication-Ready Figures</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-cheatsheet.html"><span class="progress-dot"></span>ggplot2 Quickref</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Statistics<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> EDA & Data Quality</li><li data-subkey="sec3sub1"><a href="/Automated-EDA-in-R.html"><span class="progress-dot"></span>Automated EDA</a></li><li data-subkey="sec3sub1"><a href="/Missing-Data-Visualization-in-R-naniar.html"><span class="progress-dot"></span>Missing Data Viz (naniar)</a></li><li data-subkey="sec3sub1"><a href="/Outlier-Detection-in-R.html"><span class="progress-dot"></span>Outlier Detection</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Probability</li><li data-subkey="sec3sub2"><a href="/Sample-Spaces-Events-and-Probability-Axioms-in-R-With-Monte-Carlo-Proof.html"><span class="progress-dot"></span>Probability Axioms</a></li><li data-subkey="sec3sub2"><a href="/Conditional-Probability-in-R.html"><span class="progress-dot"></span>Conditional Probability</a></li><li data-subkey="sec3sub2"><a href="/Random-Variables-in-R.html"><span class="progress-dot"></span>Random Variables</a></li><li data-subkey="sec3sub2"><a href="/Binomial-and-Poisson-Distributions-in-R.html"><span class="progress-dot"></span>Binomial vs Poisson</a></li><li data-subkey="sec3sub2"><a href="/Normal-t-F-and-Chi-Squared-Distributions-in-R.html"><span class="progress-dot"></span>Normal, t, F, Chi-Squared</a></li><li data-subkey="sec3sub2"><a href="/Central-Limit-Theorem-in-R.html"><span class="progress-dot"></span>Central Limit Theorem</a></li><li data-subkey="sec3sub2"><a href="/Sampling-Distributions-in-R.html"><span class="progress-dot"></span>Sampling Distributions</a></li><li data-subkey="sec3sub2"><a href="/Law-of-Large-Numbers-vs-CLT-in-R.html"><span class="progress-dot"></span>LLN vs CLT</a></li><li data-subkey="sec3sub2"><a href="/What-Is-Probability-Simulation-First-Intuition-in-R-Before-the-Formulas.html"><span class="progress-dot"></span>Probability (Simulation-First)</a></li><li data-subkey="sec3sub2"><a href="/Expected-Value-and-Variance-in-R.html"><span class="progress-dot"></span>Expected Value and Variance</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Inference & Estimation</li><li data-subkey="sec3sub3"><a href="/Maximum-Likelihood-Estimation-in-R.html"><span class="progress-dot"></span>Maximum Likelihood Estimation</a></li><li data-subkey="sec3sub3"><a href="/Hypothesis-Testing-in-R.html"><span class="progress-dot"></span>Hypothesis Testing</a></li><li data-subkey="sec3sub3"><a href="/Sample-Size-Planning-in-R.html"><span class="progress-dot"></span>Sample Size Planning</a></li><li data-subkey="sec3sub3"><a href="/Which-Statistical-Test-in-R.html"><span class="progress-dot"></span>Choosing the Right Test</a></li><li data-subkey="sec3sub3"><a href="/Statistical-Tests-in-R.html"><span class="progress-dot"></span>Statistical Tests</a></li><li data-subkey="sec3sub3"><a href="/Measures-of-Association-in-R.html"><span class="progress-dot"></span>Measures of Association</a></li><li data-subkey="sec3sub3"><a href="/Point-Estimation-in-R.html"><span class="progress-dot"></span>Point Estimation</a></li><li data-subkey="sec3sub3"><a href="/Confidence-Intervals-in-R.html"><span class="progress-dot"></span>Confidence Intervals</a></li><li data-subkey="sec3sub3"><a href="/Type-I-and-Type-II-Errors-in-R.html"><span class="progress-dot"></span>Type I and II Errors</a></li><li data-subkey="sec3sub3"><a href="/Statistical-Power-Analysis-in-R.html"><span class="progress-dot"></span>Power Analysis</a></li><li data-subkey="sec3sub3"><a href="/Effect-Size-in-R.html"><span class="progress-dot"></span>Effect Size</a></li><li data-subkey="sec3sub3"><a href="/t-Tests-in-R.html"><span class="progress-dot"></span>t-Tests</a></li><li data-subkey="sec3sub3"><a href="/Proportion-Tests-in-R.html"><span class="progress-dot"></span>Proportion Tests</a></li><li data-subkey="sec3sub3"><a href="/Normality-and-Variance-Tests-in-R.html"><span class="progress-dot"></span>Normality & Variance Tests</a></li><li data-subkey="sec3sub3"><a href="/Chi-Square-Tests-in-R.html"><span class="progress-dot"></span>Chi-Square Tests</a></li><li data-subkey="sec3sub3"><a href="/Wilcoxon-Mann-Whitney-and-Kruskal-Wallis-in-R.html"><span class="progress-dot"></span>Wilcoxon, Mann-Whitney & Kruskal-Wallis</a></li><li data-subkey="sec3sub3"><a href="/Multiple-Comparisons-in-R.html"><span class="progress-dot"></span>Multiple Testing Correction</a></li><li data-subkey="sec3sub3"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Hypothesis Testing Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Regression</li><li data-subkey="sec3sub4"><a href="/Linear-Regression.html"><span class="progress-dot"></span>Linear Regression</a></li><li data-subkey="sec3sub4"><a href="/Logistic-Regression-With-R.html"><span class="progress-dot"></span>Logistic Regression</a></li><li data-subkey="sec3sub4"><a href="/Variable-Selection-and-Importance-With-R.html"><span class="progress-dot"></span>Feature Selection</a></li><li data-subkey="sec3sub4"><a href="/Model-Selection-in-R.html"><span class="progress-dot"></span>Model Selection</a></li><li data-subkey="sec3sub4"><a href="/Missing-Value-Treatment-With-R.html"><span class="progress-dot"></span>Missing Value Treatment</a></li><li data-subkey="sec3sub4"><a href="/Outlier-Treatment-With-R.html"><span class="progress-dot"></span>Outlier Analysis</a></li><li data-subkey="sec3sub4"><a href="/adv-regression-models.html"><span class="progress-dot"></span>Advanced Regression Models</a></li><li data-subkey="sec3sub4"><a href="/Linear-Regression-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Linear Regression Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Reporting</li><li data-subkey="sec3sub5"><a href="/Statistical-Consulting-in-R.html"><span class="progress-dot"></span>Statistical Consulting</a></li><li data-subkey="sec3sub5"><a href="/Statistical-Report-Writing-in-R.html"><span class="progress-dot"></span>Statistical Report Writing</a></li><li data-subkey="sec3sub5"><a href="/Bootstrap-Confidence-Intervals-in-R.html"><span class="progress-dot"></span>Bootstrap Confidence Intervals</a></li><li data-subkey="sec3sub5"><a href="/Reporting-Statistics-in-R.html"><span class="progress-dot"></span>Reporting Statistics</a></li><li data-subkey="sec3sub5"><a href="/Correlation-in-R.html"><span class="progress-dot"></span>Correlation (Pearson, Spearman, Kendall)</a></li><li data-subkey="sec3sub5"><a href="/Linear-Regression-Assumptions-in-R.html"><span class="progress-dot"></span>Linear Regression Assumptions</a></li><li data-subkey="sec3sub5"><a href="/Dummy-Variables-in-R.html"><span class="progress-dot"></span>Dummy Variables in R</a></li><li data-subkey="sec3sub5"><a href="/Interaction-Effects-in-R.html"><span class="progress-dot"></span>Interaction Effects</a></li><li data-subkey="sec3sub5"><a href="/Regression-Diagnostics-in-R.html"><span class="progress-dot"></span>Regression Diagnostics</a></li><li data-subkey="sec3sub5"><a href="/Logistic-Regression-in-R.html"><span class="progress-dot"></span>Logistic Regression (glm + ROC)</a></li><li data-subkey="sec3sub5"><a href="/Variable-Selection-in-R.html"><span class="progress-dot"></span>Variable Selection</a></li><li data-subkey="sec3sub5"><a href="/Poisson-Regression-in-R.html"><span class="progress-dot"></span>Poisson Regression</a></li><li data-subkey="sec3sub5"><a href="/Ridge-and-Lasso-Regression-in-R.html"><span class="progress-dot"></span>Ridge & Lasso Regression</a></li><li data-subkey="sec3sub5"><a href="/Polynomial-and-Spline-Regression-in-R.html"><span class="progress-dot"></span>Polynomial & Splines</a></li><li data-subkey="sec3sub5"><a href="/Regression-Tables-in-R.html"><span class="progress-dot"></span>Regression Tables (3 packages)</a></li><li data-subkey="sec3sub5"><a href="/One-Way-ANOVA-in-R.html"><span class="progress-dot"></span>One-Way ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Post-Hoc-Tests-After-ANOVA.html"><span class="progress-dot"></span>Post-Hoc Tests After ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Two-Way-ANOVA-in-R.html"><span class="progress-dot"></span>Two-Way ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Repeated-Measures-ANOVA-in-R.html"><span class="progress-dot"></span>Repeated Measures ANOVA</a></li><li data-subkey="sec3sub5"><a href="/ANCOVA-in-R.html"><span class="progress-dot"></span>ANCOVA</a></li><li data-subkey="sec3sub5"><a href="/Experimental-Design-Principles-in-R.html"><span class="progress-dot"></span>Experimental Design in R</a></li><li data-subkey="sec3sub5"><a href="/Factorial-Experiments-in-R.html"><span class="progress-dot"></span>Factorial Designs (2^k)</a></li><li data-subkey="sec3sub5"><a href="/AB-Testing-in-R.html"><span class="progress-dot"></span>A/B Testing</a></li><li data-subkey="sec3sub5"><a href="/MANOVA-in-R.html"><span class="progress-dot"></span>MANOVA</a></li><li data-subkey="sec3sub5"><a href="/Mixed-ANOVA-in-R.html"><span class="progress-dot"></span>Mixed ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Multivariate-Statistics-in-R.html"><span class="progress-dot"></span>Multivariate Distances & Hotelling's T²</a></li><li data-subkey="sec3sub5"><a href="/PCA-in-R.html"><span class="progress-dot"></span>PCA with prcomp()</a></li><li data-subkey="sec3sub5"><a href="/Interpreting-PCA-Results-in-R.html"><span class="progress-dot"></span>Interpreting PCA Output</a></li><li data-subkey="sec3sub5"><a href="/Exploratory-Factor-Analysis-in-R.html"><span class="progress-dot"></span>Exploratory Factor Analysis</a></li><li 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href="/Robust-Regression-in-R.html"><span class="progress-dot"></span>Robust Regression (rlm)</a></li><li data-subkey="sec3sub5"><a href="/factoextra-and-FactoMineR.html"><span class="progress-dot"></span>factoextra (PCA + Clusters)</a></li><li data-subkey="sec3sub5"><a href="/Categorical-Data-in-R.html"><span class="progress-dot"></span>Categorical Data (Tables & Mosaic)</a></li><li data-subkey="sec3sub5"><a href="/Chi-Square-Test-of-Independence-in-R.html"><span class="progress-dot"></span>Chi-Square Test of Independence</a></li><li data-subkey="sec3sub5"><a href="/Chi-Square-Goodness-of-Fit-Test-in-R.html"><span class="progress-dot"></span>Chi-Square Goodness-of-Fit</a></li><li data-subkey="sec3sub5"><a href="/Fishers-Exact-Test-in-R.html"><span class="progress-dot"></span>Fisher's Exact Test</a></li><li data-subkey="sec3sub5"><a href="/Odds-Ratios-and-Relative-Risk-in-R.html"><span class="progress-dot"></span>Odds Ratios & Relative Risk</a></li><li data-subkey="sec3sub5"><a href="/Logistic-Regression-in-R-2.html"><span class="progress-dot"></span>Logistic Regression (Diagnostics)</a></li><li data-subkey="sec3sub5"><a href="/Poisson-and-Negative-Binomial-Regression.html"><span class="progress-dot"></span>Poisson & Negative Binomial Regression</a></li><li data-subkey="sec3sub5"><a href="/Multinomial-and-Ordinal-Logistic-Regression-in-R.html"><span class="progress-dot"></span>Multinomial & Ordinal Logistic Regression</a></li><li data-subkey="sec3sub5"><a href="/When-to-Use-Nonparametric-Tests-in-R.html"><span class="progress-dot"></span>When to Use Nonparametric Tests</a></li><li data-subkey="sec3sub5"><a href="/Wilcoxon-Signed-Rank-Test-in-R.html"><span class="progress-dot"></span>Wilcoxon Signed-Rank Test</a></li><li data-subkey="sec3sub5"><a href="/Mann-Whitney-U-Test-in-R.html"><span class="progress-dot"></span>Mann-Whitney U Test</a></li><li data-subkey="sec3sub5"><a href="/Kruskal-Wallis-Test-in-R-2.html"><span class="progress-dot"></span>Kruskal-Wallis Test</a></li><li data-subkey="sec3sub5"><a href="/Friedman-Test-in-R.html"><span class="progress-dot"></span>Friedman Test</a></li><li data-subkey="sec3sub5"><a href="/Spearman-and-Kendall-Correlation-in-R.html"><span class="progress-dot"></span>Spearman & Kendall Correlation</a></li><li data-subkey="sec3sub5"><a href="/Bootstrap-in-R.html"><span class="progress-dot"></span>Bootstrap (boot package)</a></li><li data-subkey="sec3sub5"><a href="/Quantile-Regression-in-R-2.html"><span class="progress-dot"></span>Quantile Regression</a></li><li data-subkey="sec3sub5"><a href="/Matrix-Operations-in-R.html"><span class="progress-dot"></span>Matrix Operations in R</a></li><li data-subkey="sec3sub5"><a href="/Solving-Linear-Systems-in-R.html"><span class="progress-dot"></span>Solving Linear Systems in R</a></li><li data-subkey="sec3sub5"><a href="/Eigenvalues-and-Eigenvectors-in-R.html"><span class="progress-dot"></span>Eigenvalues & Eigenvectors in R</a></li><li data-subkey="sec3sub5"><a href="/Singular-Value-Decomposition-in-R.html"><span class="progress-dot"></span>Singular Value Decomposition in R</a></li><li data-subkey="sec3sub5"><a href="/Projections-and-the-Hat-Matrix-in-R.html"><span class="progress-dot"></span>Projections & the Hat Matrix</a></li><li data-subkey="sec3sub5"><a href="/QR-Decomposition-in-R.html"><span class="progress-dot"></span>QR Decomposition in R</a></li><li data-subkey="sec3sub5"><a href="/Quadratic-Forms-in-R.html"><span class="progress-dot"></span>Quadratic Forms</a></li><li data-subkey="sec3sub5"><a href="/Matrix-Derivatives-and-the-Hessian-in-R.html"><span class="progress-dot"></span>Matrix Derivatives & Hessian</a></li><li data-subkey="sec3sub5"><a href="/Exponential-Family-Distributions-in-R.html"><span class="progress-dot"></span>Exponential Family Distributions</a></li><li data-subkey="sec3sub5"><a href="/Sufficient-Statistics-in-R.html"><span class="progress-dot"></span>Sufficient Statistics</a></li><li data-subkey="sec3sub5"><a href="/Complete-and-Ancillary-Statistics-in-R.html"><span class="progress-dot"></span>Complete & Ancillary Statistics</a></li><li data-subkey="sec3sub5"><a href="/UMVUE-in-R-2.html"><span class="progress-dot"></span>UMVUE (Rao-Blackwell & Lehmann-Scheffé)</a></li><li data-subkey="sec3sub5"><a href="/Cramer-Rao-Lower-Bound-in-R-2.html"><span class="progress-dot"></span>Cramér-Rao Lower Bound</a></li><li data-subkey="sec3sub5"><a href="/Asymptotic-Theory-in-R-2.html"><span class="progress-dot"></span>Asymptotic Theory</a></li><li data-subkey="sec3sub5"><a href="/Neyman-Pearson-Lemma-in-R-2.html"><span class="progress-dot"></span>Neyman-Pearson Lemma</a></li><li data-subkey="sec3sub5"><a href="/Likelihood-Ratio-Tests-and-Pivotal-Methods.html"><span class="progress-dot"></span>Likelihood Ratio & Pivotal Methods</a></li><li data-subkey="sec3sub5"><a href="/Decision-Theory-in-R.html"><span class="progress-dot"></span>Decision Theory</a></li><li data-subkey="sec3sub5"><a href="/Asymptotic-Relative-Efficiency-in-R.html"><span class="progress-dot"></span>Asymptotic Relative Efficiency</a></li><li data-subkey="sec3sub5"><a href="/Bayes-Theorem-in-R.html"><span class="progress-dot"></span>Bayes' Theorem</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Statistics-in-R.html"><span class="progress-dot"></span>Bayesian Statistics</a></li><li data-subkey="sec3sub5"><a href="/Conjugate-Priors-in-R.html"><span class="progress-dot"></span>Conjugate Priors</a></li><li data-subkey="sec3sub5"><a href="/Grid-Approximation-in-R.html"><span class="progress-dot"></span>Grid Approximation</a></li><li data-subkey="sec3sub5"><a href="/MCMC-in-R.html"><span class="progress-dot"></span>MCMC in R</a></li><li data-subkey="sec3sub5"><a href="/Gibbs-Sampling-in-R.html"><span class="progress-dot"></span>Gibbs Sampling</a></li><li data-subkey="sec3sub5"><a href="/Hamiltonian-Monte-Carlo-in-R.html"><span class="progress-dot"></span>Hamiltonian Monte Carlo</a></li><li data-subkey="sec3sub5"><a href="/Stan-in-R.html"><span class="progress-dot"></span>Stan</a></li><li data-subkey="sec3sub5"><a href="/brms-in-R.html"><span class="progress-dot"></span>brms</a></li><li data-subkey="sec3sub5"><a href="/Choosing-Priors-in-R.html"><span class="progress-dot"></span>Choosing Priors</a></li><li data-subkey="sec3sub5"><a href="/Prior-Predictive-Checks-in-R.html"><span class="progress-dot"></span>Prior Predictive Checks</a></li><li data-subkey="sec3sub5"><a href="/Compare-Bayesian-Models-in-R.html"><span class="progress-dot"></span>Compare Bayesian Models</a></li><li data-subkey="sec3sub5"><a href="/Posterior-Predictive-Checks-in-R.html"><span class="progress-dot"></span>Posterior Predictive Checks</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Linear-Regression-in-R.html"><span class="progress-dot"></span>Bayesian Linear Regression</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Logistic-Regression-in-R.html"><span class="progress-dot"></span>Bayesian Logistic Regression</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Hierarchical-Models-in-R.html"><span class="progress-dot"></span>Bayesian Hierarchical Models</a></li><li data-subkey="sec3sub5"><a href="/Multilevel-Models-in-R.html"><span class="progress-dot"></span>Multilevel Models</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-ANOVA-in-R.html"><span class="progress-dot"></span>Bayesian ANOVA</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Machine Learning</li><li data-subkey="sec3sub6"><a href="/Machine-Learning-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Machine Learning Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Time Series<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li data-subkey="sec4sub0"><a href="/Time-Series-Analysis-With-R.html"><span class="progress-dot"></span>Time Series Analysis</a></li><li data-subkey="sec4sub0"><a href="/Time-Series-Forecasting-With-R.html"><span class="progress-dot"></span>Time Series Forecasting</a></li><li data-subkey="sec4sub0"><a href="/Time-Series-Forecasting-With-R-part2.html"><span class="progress-dot"></span>More Time Series Forecasting</a></li><li data-subkey="sec4sub0"><a href="/Time-Series-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Time Series Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Advanced R<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Functional Programming</li><li data-subkey="sec5sub1"><a href="/Functional-Programming-in-R.html"><span class="progress-dot"></span>Functional Programming</a></li><li data-subkey="sec5sub1"><a href="/R-Functional-Programming-Exercises-quiz.html"><span class="progress-dot"></span>Functional Programming Quiz</a></li><li data-subkey="sec5sub1"><a href="/purrr-map-Variants.html"><span class="progress-dot"></span>purrr map() Variants</a></li><li data-subkey="sec5sub1"><a href="/R-Anonymous-Functions.html"><span class="progress-dot"></span>R Anonymous Functions</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Factories.html"><span class="progress-dot"></span>R Function Factories</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Operators.html"><span class="progress-dot"></span>R Function Operators</a></li><li data-subkey="sec5sub1"><a href="/Reduce-Filter-Map-in-R.html"><span class="progress-dot"></span>Reduce, Filter, Map</a></li><li data-subkey="sec5sub1"><a href="/Memoization-in-R.html"><span class="progress-dot"></span>Memoization in R</a></li><li data-subkey="sec5sub1"><a href="/Writing-Composable-R-Code.html"><span class="progress-dot"></span>Composable R Code</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> OOP in R</li><li data-subkey="sec5sub2"><a href="/OOP-in-R.html"><span class="progress-dot"></span>OOP in R: S3/S4/R6</a></li><li data-subkey="sec5sub2"><a href="/S3-Classes-in-R.html"><span class="progress-dot"></span>S3 Classes</a></li><li data-subkey="sec5sub2"><a href="/S3-Method-Dispatch-in-R.html"><span class="progress-dot"></span>S3 Method Dispatch</a></li><li data-subkey="sec5sub2"><a href="/S4-Classes-in-R.html"><span class="progress-dot"></span>S4 Classes</a></li><li data-subkey="sec5sub2"><a href="/S4-Methods-in-R.html"><span class="progress-dot"></span>S4 Methods & Dispatch</a></li><li data-subkey="sec5sub2"><a href="/R6-Classes-in-R.html"><span class="progress-dot"></span>R6 Classes</a></li><li data-subkey="sec5sub2"><a href="/R6-Advanced.html"><span class="progress-dot"></span>R6 Advanced</a></li><li data-subkey="sec5sub2"><a href="/Operator-Overloading-in-R.html"><span class="progress-dot"></span>Operator Overloading</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> How R Works</li><li data-subkey="sec5sub3"><a href="/R-Names-and-Values.html"><span class="progress-dot"></span>R Names & Values</a></li><li data-subkey="sec5sub3"><a href="/R-Assignment-Deep-Dive.html"><span class="progress-dot"></span>R Assignment Deep Dive</a></li><li data-subkey="sec5sub3"><a href="/R-Memory-lobstr.html"><span class="progress-dot"></span>R Memory & lobstr</a></li><li data-subkey="sec5sub3"><a href="/R-Environments.html"><span class="progress-dot"></span>R Environments</a></li><li data-subkey="sec5sub3"><a href="/R-Lexical-Scoping.html"><span class="progress-dot"></span>Lexical Scoping</a></li><li data-subkey="sec5sub3"><a href="/R-Closures.html"><span class="progress-dot"></span>R Closures</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Debugging & Performance</li><li data-subkey="sec5sub4"><a href="/R-Conditions-System.html"><span class="progress-dot"></span>Conditions System</a></li><li data-subkey="sec5sub4"><a href="/R-Debugging.html"><span class="progress-dot"></span>Debugging R Code</a></li><li data-subkey="sec5sub4"><a href="/R-Common-Errors.html"><span class="progress-dot"></span>50 Common R Errors</a></li><li data-subkey="sec5sub4"><a href="/Parallel-Computing-With-R.html"><span class="progress-dot"></span>Parallel Computing</a></li><li data-subkey="sec5sub4"><a href="/Strategies-To-Improve-And-Speedup-R-Code.html"><span class="progress-dot"></span>Speedup R Code</a></li><li data-subkey="sec5sub4"><a href="/Shiny-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Shiny Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Classic Tutorials<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li data-subkey="sec6sub0"><a href="/R-Tutorial.html"><span class="progress-dot"></span>R Tutorial (Classic)</a></li><li data-subkey="sec6sub0"><a href="/ggplot2-Tutorial-With-R.html"><span class="progress-dot"></span>ggplot2 Short Tutorial</a></li><li data-subkey="sec6sub0"><a href="/Complete-Ggplot2-Tutorial-Part1-With-R-Code.html"><span class="progress-dot"></span>ggplot2 Tutorial 1 - Intro</a></li><li data-subkey="sec6sub0"><a href="/Complete-Ggplot2-Tutorial-Part2-Customizing-Theme-With-R-Code.html"><span class="progress-dot"></span>ggplot2 Tutorial 2 - Theme</a></li><li data-subkey="sec6sub0"><a href="/Top50-Ggplot2-Visualizations-MasterList-R-Code.html"><span class="progress-dot"></span>ggplot2 Tutorial 3 - Masterlist</a></li><li data-subkey="sec6sub0"><a href="/Association-Mining-With-R.html"><span class="progress-dot"></span>Association Mining</a></li><li data-subkey="sec6sub0"><a href="/Multi-Dimensional-Scaling-With-R.html"><span class="progress-dot"></span>Multi Dimensional Scaling</a></li><li data-subkey="sec6sub0"><a href="/Optimization-With-R.html"><span class="progress-dot"></span>Optimization</a></li><li data-subkey="sec6sub0"><a href="/Information-Value-With-R.html"><span class="progress-dot"></span>InformationValue Package</a></li></ul></li><li class="sidebar-section expanded"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Practice Exercises<span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Mastery Quizzes (Certificate)</li><li data-subkey="sec7sub1"><a href="/R-Beginner-Exercises-quiz.html"><span class="progress-dot"></span>R Fundamentals Quiz</a></li><li data-subkey="sec7sub1"><a href="/dplyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>dplyr Quiz</a></li><li data-subkey="sec7sub1"><a href="/ggplot2-Exercises-in-R-quiz.html"><span class="progress-dot"></span>ggplot2 Quiz</a></li><li data-subkey="sec7sub1"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Hypothesis Testing Quiz</a></li><li data-subkey="sec7sub1"><a href="/Linear-Regression-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Linear Regression Quiz</a></li><li data-subkey="sec7sub1"><a href="/Machine-Learning-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Machine Learning Quiz</a></li><li data-subkey="sec7sub1"><a href="/tidyr-Exercises-in-R-quiz.html"><span 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<h1>Bayesian Statistics Exercises in R: 20 Practice Problems</h1>
<p class="lead">Twenty <a class="auto-link" href="Bayesian-Statistics-in-R.html" title="Bayesian Statistics in R: Build Genuine Intuition Before Opening Stan or brms">Bayesian statistics</a> practice problems in R covering <a class="auto-link" href="Gamma-and-Beta-Distributions-in-R.html" title="Gamma & Beta Distributions in R: Shape, Scale & Conjugate Priors">conjugate priors</a>, <a class="auto-link" href="Grid-Approximation-in-R.html" title="Grid Approximation in R: Compute Bayesian Posteriors Without MCMC">grid approximation</a>, posterior summaries, regression with rstanarm and brms, MCMC diagnostics (R-hat, ESS), posterior predictive checks, credible intervals, and LOO model comparison. Solutions are hidden; reveal each one only after you have attempted the exercise yourself on a local R install.</p>
<div class="post-byline" style="color:#6b7280;font-size:14px;margin:2px 0 18px 0;line-height:1.5;">By <strong>Selva Prabhakaran</strong> · Published May 17, 2026 · Last updated May 17, 2026</div>
<div class="engagement-header" data-difficulty="Mixed" data-time="45" data-exercises="20" data-xp="300"></div>
<pre><code class="language-r">library(rstanarm)
library(brms)
library(bayesplot)
library(posterior)
library(bayestestR)
library(loo)</code></pre>
<h2>Section 1. Conjugate updating and grid approximation (4 problems)</h2>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-1-1" data-grade-mode="output-compare" data-difficulty="beginner">
<h3 class="exercise-title">Exercise 1.1: Beta-binomial conjugate update for an A/B test</h3>
<p class="exercise-task"><strong>Task:</strong> A growth team is running an A/B test on a landing page. The control variant gets 14 conversions out of 100 visitors. Use a Beta(2, 2) <a class="auto-link" href="Choosing-Priors-in-R.html" title="Choosing Priors in R: Why Your Bayesian Result Depends on This One Decision">weakly informative prior</a> on the conversion rate and compute the resulting Beta posterior parameters analytically. Save the posterior shape parameters as a length-2 numeric vector to <code>ex_1_1</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> [1] 16 88</code></pre>
</div>
<p><strong>Difficulty:</strong> Beginner</p>
<div class="exercise-hints" hidden><p>A Beta prior updated with binomial data stays a Beta; its two new shape parameters are the old ones with each outcome's count folded in.</p><p>Add the 14 successes to the first prior parameter and the 100 - 14 failures to the second, then bundle both with <code>c()</code>.</p></div>
<pre><code class="language-r">ex_1_1 <- # your code here
ex_1_1</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">prior_a <- 2
prior_b <- 2
successes <- 14
failures <- 100 - successes
ex_1_1 <- c(prior_a + successes, prior_b + failures)
ex_1_1
#> [1] 16 88</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> When the prior on a proportion is Beta(a, b) and the data are binomial with k successes out of n trials, conjugacy gives a closed-form Beta(a+k, b+n-k) posterior. Beta(2, 2) is a soft prior centered at 0.5 with an equivalent weight of about 2 successes plus 2 failures. The resulting Beta(16, 88) posterior shifts strongly toward 0.16 because the data dominate the prior at n=100. This identity is the foundation for streaming A/B updates without refitting.</p>
</details>
</section>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-1-2" data-grade-mode="output-compare" data-difficulty="intermediate">
<h3 class="exercise-title">Exercise 1.2: Posterior mean and 95% credible interval from Beta(16, 88)</h3>
<p class="exercise-task"><strong>Task:</strong> Continuing the A/B test from the previous exercise, summarize the Beta(16, 88) posterior by computing the posterior mean conversion rate and the equal-tailed 95% <a class="auto-link" href="Bayesian-Linear-Regression-in-R.html" title="Bayesian Linear Regression in R: Get Uncertainty Estimates lm() Cannot Give You">credible interval</a> using base R quantile functions. Save a named numeric vector with components <code>mean</code>, <code>lower</code>, <code>upper</code> to <code>ex_1_2</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> mean lower upper
#> 0.1538462 0.0935023 0.2256617</code></pre>
</div>
<p><strong>Difficulty:</strong> Intermediate</p>
<div class="exercise-hints" hidden><p>The mean of a Beta is its first parameter divided by the sum of both, and the interval endpoints are just posterior quantiles.</p><p>Compute <code>16 / (16 + 88)</code> for the mean and <code>qbeta(c(0.025, 0.975), 16, 88)</code> for the bounds, then name them with <code>c(mean = ..., lower = ..., upper = ...)</code>.</p></div>
<pre><code class="language-r">ex_1_2 <- # your code here
ex_1_2</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">post_a <- 16
post_b <- 88
post_mean <- post_a / (post_a + post_b)
ci <- qbeta(c(0.025, 0.975), post_a, post_b)
ex_1_2 <- c(mean = post_mean, lower = ci[1], upper = ci[2])
ex_1_2
#> mean lower upper
#> 0.1538462 0.0935023 0.2256617</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> The mean of Beta(a, b) is a / (a + b), and equal-tailed quantiles come straight from <code>qbeta()</code>. The 95% credible interval is a direct probability statement about the parameter: there is a 95% <a class="auto-link" href="Bayes-Theorem-in-R.html" title="Bayes' Theorem in R: Why Medical Tests Mislead You, A Simulation That Shows Why">posterior probability</a> the true conversion rate lies between 9.4% and 22.6%, unlike a frequentist confidence interval which is a statement about the procedure. A highest-density interval would be slightly narrower because the Beta posterior is right-skewed.</p>
</details>
</section>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-1-3" data-grade-mode="output-compare" data-difficulty="intermediate">
<h3 class="exercise-title">Exercise 1.3: Gamma-Poisson conjugate update for a call center rate</h3>
<p class="exercise-task"><strong>Task:</strong> A call center operations analyst observes 120 calls arriving in 8 hours and assumes Poisson arrivals with rate <code>lambda</code> per hour. With a weakly informative Gamma(2, 1) prior on <code>lambda</code>, compute the analytical Gamma posterior parameters (shape and rate) and the posterior mean rate. Save a named numeric vector with components <code>shape</code>, <code>rate</code>, <code>mean</code> to <code>ex_1_3</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> shape rate mean
#> 122.000000 9.000000 13.555556</code></pre>
</div>
<p><strong>Difficulty:</strong> Intermediate</p>
<div class="exercise-hints" hidden><p>A Gamma prior on a Poisson rate updates by accumulating total counts and total exposure time onto its two parameters.</p><p>Set the posterior shape to <code>prior_shape + 120</code> and the posterior rate to <code>prior_rate + 8</code>, then the mean is shape divided by rate.</p></div>
<pre><code class="language-r">ex_1_3 <- # your code here
ex_1_3</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">prior_shape <- 2
prior_rate <- 1
total_calls <- 120
hours <- 8
post_shape <- prior_shape + total_calls
post_rate <- prior_rate + hours
post_mean <- post_shape / post_rate
ex_1_3 <- c(shape = post_shape, rate = post_rate, mean = post_mean)
ex_1_3
#> shape rate mean
#> 122.000000 9.000000 13.555556</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> For Poisson observations with a Gamma(alpha, beta) prior on the rate, the posterior is Gamma(alpha + total counts, beta + total exposure time), so the Gamma is conjugate to the Poisson and updating is just adding <a class="auto-link" href="Sufficiency-in-Statistics.html" title="Sufficiency in Statistics in R: Sufficient Statistics, Fisher-Neyman Factorization">sufficient statistics</a>. The posterior mean of 122/9 (about 13.56 calls per hour) is a precision-weighted average of the prior mean (2) and the empirical rate (15), tilted toward the data because the prior carries only 1 hour of equivalent weight versus 8 hours of observation.</p>
</details>
</section>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-1-4" data-grade-mode="output-compare" data-difficulty="intermediate">
<h3 class="exercise-title">Exercise 1.4: Grid approximation of a Bayesian binomial proportion</h3>
<p class="exercise-task"><strong>Task:</strong> A biostatistician wants to estimate the cure rate after observing 7 successes in 10 patients, without using closed-form conjugacy. Approximate the posterior over a 1,000-point uniform grid on [0, 1] with a uniform prior and a binomial likelihood, then compute the posterior mean by a weighted sum of grid values. Save the result as a single numeric value to <code>ex_1_4</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> [1] 0.6666007</code></pre>
</div>
<p><strong>Difficulty:</strong> Intermediate</p>
<div class="exercise-hints" hidden><p>Evaluate prior times likelihood across a fine grid, normalize the weights to sum to one, then take a weighted average of the grid values.</p><p>Build the grid with <code>seq(0, 1, length.out = 1000)</code>, get the likelihood from <code>dbinom(7, size = 10, prob = p_grid)</code>, and finish with <code>sum(p_grid * post)</code>.</p></div>
<pre><code class="language-r">ex_1_4 <- # your code here
ex_1_4</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">p_grid <- seq(0, 1, length.out = 1000)
prior <- rep(1, 1000)
lik <- dbinom(7, size = 10, prob = p_grid)
post <- (lik * prior) / sum(lik * prior)
ex_1_4 <- sum(p_grid * post)
ex_1_4
#> [1] 0.6666007</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> Grid approximation discretizes the parameter space, evaluates prior times likelihood at each grid point, and normalizes the weights to sum to one. The posterior mean is then a weighted average of the grid values. The exact Beta(8, 4) posterior under a uniform prior has mean 8/12 = 0.6667, so the 1,000-point grid agrees to four decimals. Grid approximation breaks down past three or four parameters because the grid size grows exponentially with dimensionality, which is why MCMC takes over.</p>
</details>
</section>
<h2>Section 2. Bayesian regression with rstanarm (4 problems)</h2>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-2-1" data-grade-mode="output-compare" data-difficulty="intermediate">
<h3 class="exercise-title">Exercise 2.1: Bayesian linear regression on mtcars with stan_glm</h3>
<p class="exercise-task"><strong>Task:</strong> An automotive performance analyst wants to model <code>mpg</code> as a linear function of <code>wt</code> on the built-in <code>mtcars</code> dataset using <code>rstanarm::stan_glm()</code> with default weakly informative priors, 2,000 iterations, and 4 chains. Fit the model with <code>seed = 1</code> for reproducibility and save the fitted model object to <code>ex_2_1</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> stan_glm
#> family: gaussian [identity]
#> formula: mpg ~ wt
#> observations: 32
#> Median MAD_SD
#> (Intercept) 37.3 1.9
#> wt -5.3 0.6
#> sigma 3.2 0.4</code></pre>
</div>
<p><strong>Difficulty:</strong> Intermediate</p>
<div class="exercise-hints" hidden><p>Fit the regression with a sampler that returns full posterior draws rather than a single least-squares point estimate.</p><p>Call <code>rstanarm::stan_glm(mpg ~ wt, data = mtcars, family = gaussian(), iter = 2000, chains = 4, seed = 1)</code>.</p></div>
<pre><code class="language-r">ex_2_1 <- # your code here
ex_2_1</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">ex_2_1 <- rstanarm::stan_glm(
mpg ~ wt,
data = mtcars,
family = gaussian(),
iter = 2000,
chains = 4,
refresh = 0,
seed = 1
)
print(ex_2_1)
#> stan_glm
#> family: gaussian [identity]
#> formula: mpg ~ wt
#> Median MAD_SD
#> (Intercept) 37.3 1.9
#> wt -5.3 0.6
#> sigma 3.2 0.4</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> <code>stan_glm()</code> is the rstanarm wrapper around Stan that fits the same families as base <code>glm()</code> but returns full posterior draws rather than maximum likelihood estimates. Default rstanarm priors are weakly informative normals centered at zero with scales auto-adjusted to the data, so for moderate sample sizes the posterior is essentially data-driven. The printed output shows posterior medians and median absolute deviations rather than point estimates plus standard errors. Set <code>refresh = 0</code> to suppress chain progress messages in batch scripts.</p>
</details>
</section>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-2-2" data-grade-mode="output-compare" data-difficulty="advanced">
<h3 class="exercise-title">Exercise 2.2: Set custom normal priors on regression coefficients</h3>
<p class="exercise-task"><strong>Task:</strong> The analyst from the previous exercise wants to enforce a domain-informed prior belief that the slope of <code>mpg</code> on <code>wt</code> is around -5 with standard deviation 2, and a normal(20, 2.5) prior on the centered intercept. Refit the same model with these priors using <code>rstanarm::stan_glm()</code> and save the fitted object to <code>ex_2_2</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> Priors for model 'ex_2_2'
#> ------
#> Intercept (after predictors centered)
#> ~ normal(location = 20, scale = 2.5)
#> Coefficients
#> ~ normal(location = -5, scale = 2)
#> Auxiliary (sigma)
#> ~ exponential(rate = 0.31)</code></pre>
</div>
<p><strong>Difficulty:</strong> Advanced</p>
<div class="exercise-hints" hidden><p>Override the default coefficient and intercept priors by handing your own distributions to the fitting call.</p><p>Pass <code>prior = rstanarm::normal(location = -5, scale = 2)</code> and <code>prior_intercept = rstanarm::normal(location = 20, scale = 2.5)</code> to <code>stan_glm()</code>.</p></div>
<pre><code class="language-r">ex_2_2 <- # your code here
ex_2_2</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">ex_2_2 <- rstanarm::stan_glm(
mpg ~ wt,
data = mtcars,
family = gaussian(),
prior = rstanarm::normal(location = -5, scale = 2),
prior_intercept = rstanarm::normal(location = 20, scale = 2.5),
iter = 2000,
chains = 4,
refresh = 0,
seed = 1
)
rstanarm::prior_summary(ex_2_2)</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> The <code>prior</code> argument controls the prior on slope coefficients while <code>prior_intercept</code> controls the prior on the centered intercept; rstanarm internally centers predictors before sampling for efficiency, so the intercept prior is on that centered intercept, not the raw one. <code>rstanarm::normal()</code> is the prior constructor and accepts vector arguments when you want a different normal per coefficient. Always confirm with <code>prior_summary()</code> because rstanarm sometimes rescales user priors when <code>autoscale = TRUE</code>.</p>
</details>
</section>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-2-3" data-grade-mode="output-compare" data-difficulty="advanced">
<h3 class="exercise-title">Exercise 2.3: Bayesian logistic regression for transmission type</h3>
<p class="exercise-task"><strong>Task:</strong> A vintage car valuation team wants the posterior over the log-odds that a car has a manual transmission (<code>am = 1</code>) as a function of <code>mpg</code> and <code>hp</code> from <code>mtcars</code>. Fit a <a class="auto-link" href="Bayesian-Logistic-Regression-in-R.html" title="Bayesian Logistic Regression in R: Why Stakeholders Trust These Odds Ratios More">Bayesian logistic regression</a> with <code>rstanarm::stan_glm()</code>, <code>family = binomial(link = "logit")</code>, 2,000 iterations, and 4 chains. Save the fitted model object to <code>ex_2_3</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> stan_glm
#> family: binomial [logit]
#> formula: am ~ mpg + hp
#> observations: 32
#> Median MAD_SD
#> (Intercept) -33.6 19.1
#> mpg 1.5 0.8
#> hp 0.05 0.04</code></pre>
</div>
<p><strong>Difficulty:</strong> Advanced</p>
<div class="exercise-hints" hidden><p>Model the binary transmission outcome on the log-odds scale by choosing the <a class="auto-link" href="Logistic-Regression-in-R.html" title="Logistic Regression in R: From glm() to Odds Ratios, ROC, and AUC">binomial family</a> for the Bayesian fit.</p><p>Call <code>rstanarm::stan_glm(am ~ mpg + hp, data = mtcars, family = binomial(link = "logit"), iter = 2000, chains = 4, seed = 1)</code>.</p></div>
<pre><code class="language-r">ex_2_3 <- # your code here
ex_2_3</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">ex_2_3 <- rstanarm::stan_glm(
am ~ mpg + hp,
data = mtcars,
family = binomial(link = "logit"),
iter = 2000,
chains = 4,
refresh = 0,
seed = 1
)
print(ex_2_3)</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> For a binomial family, <code>stan_glm()</code> samples directly from the posterior of the log-odds coefficients, so the reported medians are interpretable as posterior median log-odds-ratios per unit change in the predictor. The <code>MAD_SD</code> column reports median absolute deviation, a robust scale measure that is more reliable than the posterior standard deviation when the posterior is heavy-tailed. With only 32 observations the intercept posterior is very wide; the Bayesian framing carries that uncertainty forward into predictions instead of pretending it does not exist.</p>
</details>
</section>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-2-4" data-grade-mode="output-compare" data-difficulty="intermediate">
<h3 class="exercise-title">Exercise 2.4: Extract 90% credible intervals from a fitted stan_glm</h3>
<p class="exercise-task"><strong>Task:</strong> A reporting analyst needs to publish 90% credible intervals for every coefficient in the <a class="auto-link" href="Linear-Regression.html" title="Linear Regression">linear regression</a> <code>ex_2_1</code>. Use <code>rstanarm::posterior_interval()</code> with <code>prob = 0.90</code> to extract the matrix of intervals and save the matrix (with rows for each parameter) to <code>ex_2_4</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> 5% 95%
#> (Intercept) 34.1438 40.5113
#> wt -6.3084 -4.3110
#> sigma 2.6051 3.9523</code></pre>
</div>
<p><strong>Difficulty:</strong> Intermediate</p>
<div class="exercise-hints" hidden><p>Pull equal-tailed intervals straight from the stored posterior draws of the already-fitted model.</p><p>Call <code>rstanarm::posterior_interval()</code> on <code>ex_2_1</code> with <code>prob = 0.90</code>.</p></div>
<pre><code class="language-r">ex_2_4 <- # your code here
ex_2_4</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">ex_2_4 <- rstanarm::posterior_interval(ex_2_1, prob = 0.90)
ex_2_4
#> 5% 95%
#> (Intercept) 34.1438 40.5113
#> wt -6.3084 -4.3110
#> sigma 2.6051 3.9523</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> <code>posterior_interval()</code> extracts equal-tailed credible intervals from the stored MCMC draws. The default <code>prob = 0.9</code> is the rstanarm convention because the authors argue 95% intervals overstate certainty in noisy regression scenarios. For a highest-density interval instead of equal-tailed bounds, use <code>bayestestR::hdi(ex_2_1, ci = 0.9)</code>, which gives narrower bounds when the posterior is skewed (commonly for <a class="auto-link" href="Multilevel-Models-in-R.html" title="Multilevel Models in R With brms: When Group Differences Beat Group Averages">variance components</a> or rare-event probabilities).</p>
</details>
</section>
<h2>Section 3. Bayesian regression with brms (4 problems)</h2>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-3-1" data-grade-mode="output-compare" data-difficulty="advanced">
<h3 class="exercise-title">Exercise 3.1: brm linear regression with an informative prior</h3>
<p class="exercise-task"><strong>Task:</strong> A racing engineer treating <code>mtcars</code> as historical telemetry believes the slope of <code>mpg</code> on <code>wt</code> lies around -4 with standard deviation 1 based on prior aerodynamic studies. Fit <code>brms::brm(mpg ~ wt, data = mtcars)</code> with a <code>normal(-4, 1)</code> prior on the <code>wt</code> coefficient, 4 chains, 2,000 iterations, and <code>seed = 1</code>. Save the fitted brmsfit object to <code>ex_3_1</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> Family: gaussian
#> Links: mu = identity; sigma = identity
#> Formula: mpg ~ wt
#> Data: mtcars (Number of observations: 32)
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept 35.79 1.71 32.43 39.16 1.00 3201 2814
#> wt -4.92 0.53 -5.99 -3.89 1.00 3252 2832</code></pre>
</div>
<p><strong>Difficulty:</strong> Advanced</p>
<div class="exercise-hints" hidden><p>Fit the regression with brms and attach a domain-informed prior to just the weight coefficient.</p><p>Pass <code>prior = brms::prior(normal(-4, 1), class = "b", coef = "wt")</code> to <code>brms::brm(mpg ~ wt, data = mtcars, chains = 4, iter = 2000, seed = 1)</code>.</p></div>
<pre><code class="language-r">ex_3_1 <- # your code here
ex_3_1</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">ex_3_1 <- brms::brm(
mpg ~ wt,
data = mtcars,
prior = brms::prior(normal(-4, 1), class = "b", coef = "wt"),
chains = 4,
iter = 2000,
refresh = 0,
seed = 1
)
summary(ex_3_1)</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> brms compiles a Stan program from the R formula and runs <a class="auto-link" href="Hamiltonian-Monte-Carlo-in-R.html" title="Hamiltonian Monte Carlo in R: The Physics Trick That Makes Stan So Fast">Hamiltonian Monte Carlo</a>. Priors are declared with <code>brms::prior()</code> keyed by class (<code>b</code> for coefficients, <code>Intercept</code> for the centered intercept, <code>sigma</code> for residual SD). When the prior conflicts with the data, the posterior compromises between them weighted by their precision; here Bayes pulls the OLS slope of about -5.3 toward the prior mean of -4 because n = 32 carries only moderate evidence. Run a prior sensitivity sweep before publishing.</p>
</details>
</section>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-3-2" data-grade-mode="output-compare" data-difficulty="advanced">
<h3 class="exercise-title">Exercise 3.2: Hierarchical model with random intercepts by cylinder group</h3>
<p class="exercise-task"><strong>Task:</strong> A motorsport analytics team wants to partial-pool the intercept of <code>mpg ~ wt</code> across the three cylinder groups in <code>mtcars</code> because cars with the same cylinder count share an unobserved efficiency baseline. Fit <code>mpg ~ wt + (1 | cyl)</code> with <code>brms::brm()</code> using default priors, 4 chains, 2,000 iterations, and <code>seed = 1</code>. Save the fitted brmsfit object to <code>ex_3_2</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> Multilevel Hyperparameters:
#> ~cyl (Number of levels: 3)
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(Intercept) 2.79 1.92 0.62 7.99 1.00 995 1331
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept 35.34 2.39 30.46 40.10 1.00 1893 2103
#> wt -3.97 0.71 -5.36 -2.59 1.00 2042 2087</code></pre>
</div>
<p><strong>Difficulty:</strong> Advanced</p>
<div class="exercise-hints" hidden><p>Let each cylinder group carry its own intercept drawn from a shared distribution so sparse groups borrow strength from the rest.</p><p>Use the formula <code>mpg ~ wt + (1 | cyl)</code> inside <code>brms::brm()</code> with default priors, 4 chains, and 2,000 iterations.</p></div>
<pre><code class="language-r">ex_3_2 <- # your code here
ex_3_2</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">ex_3_2 <- brms::brm(
mpg ~ wt + (1 | cyl),
data = mtcars,
chains = 4,
iter = 2000,
refresh = 0,
seed = 1
)
summary(ex_3_2)</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> The <code>(1 | cyl)</code> term tells brms to estimate one varying intercept per cylinder group, drawn from a shared normal prior whose standard deviation <code>sd(Intercept)</code> is itself estimated from the data. This is <a class="auto-link" href="Bayesian-Hierarchical-Models-in-R.html" title="Bayesian Hierarchical Models in R: The Trick That Borrows Strength Across Groups">partial pooling</a>: a group with few cars is shrunk toward the global mean, while a group with many cars stays close to its own data. Use <code>ranef(ex_3_2)</code> to inspect the per-group intercept deviations; with only three groups the partial-pooling shrinkage is strong because the group-level variance is poorly identified.</p>
</details>
</section>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-3-3" data-grade-mode="output-compare" data-difficulty="advanced">
<h3 class="exercise-title">Exercise 3.3: Compare posterior under a tight prior vs a wide prior</h3>
<p class="exercise-task"><strong>Task:</strong> A statistics teaching team wants to demonstrate how a tight prior pulls the posterior away from the OLS estimate. Refit <code>mpg ~ wt</code> on <code>mtcars</code> with <code>brms::brm()</code> twice: once with a tight <code>normal(0, 0.1)</code> prior on the slope and once with a wide <code>normal(0, 100)</code> prior. Extract the posterior median slope from each fit and save the two medians as a named numeric vector with names <code>tight</code> and <code>wide</code> to <code>ex_3_3</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> tight wide
#> -0.19030 -5.34218</code></pre>
</div>
<p><strong>Difficulty:</strong> Advanced</p>
<div class="exercise-hints" hidden><p>Fit the same model twice under priors of very different width, then read the central slope value from each posterior.</p><p>Use <code>brms::prior(normal(0, 0.1), class = "b")</code> and <code>brms::prior(normal(0, 100), class = "b")</code>, then take <code>median()</code> of <code>brms::as_draws_df(fit)$b_wt</code> for each.</p></div>
<pre><code class="language-r">ex_3_3 <- # your code here
ex_3_3</code></pre>
<details class="exercise-solution">
<summary>Click to reveal solution</summary>
<pre><code class="language-r">fit_tight <- brms::brm(
mpg ~ wt, data = mtcars,
prior = brms::prior(normal(0, 0.1), class = "b"),
chains = 4, iter = 2000, refresh = 0, seed = 1
)
fit_wide <- brms::brm(
mpg ~ wt, data = mtcars,
prior = brms::prior(normal(0, 100), class = "b"),
chains = 4, iter = 2000, refresh = 0, seed = 1
)
ex_3_3 <- c(
tight = median(brms::as_draws_df(fit_tight)$b_wt),
wide = median(brms::as_draws_df(fit_wide)$b_wt)
)
ex_3_3
#> tight wide
#> -0.19030 -5.34218</code></pre>
<p class="exercise-explanation"><strong>Explanation:</strong> With only 32 observations the data information is finite, so a tight prior centered at zero with scale 0.1 dominates and shrinks the slope toward zero, while the diffuse <code>normal(0, 100)</code> prior recovers essentially the OLS slope. This is the textbook bias-variance trade-off of prior choice: tight priors reduce posterior variance but introduce bias toward the prior mean. Always run a prior <a class="auto-link" href="Sensitivity-Analysis-in-R.html" title="Sensitivity Analysis in R: How Robust Are Your Statistical Conclusions?">sensitivity analysis</a> like this before publishing Bayesian results in any regulated reporting setting.</p>
</details>
</section>
<section class="exercise" data-exercise-id="Bayesian-Statistics-Exercises-in-R-ex-3-4" data-grade-mode="output-compare" data-difficulty="intermediate">
<h3 class="exercise-title">Exercise 3.4: Inspect the prior summary on a fitted brms model</h3>
<p class="exercise-task"><strong>Task:</strong> A reviewer wants to audit which priors brms actually used for the hierarchical fit <code>ex_3_2</code>, including the default priors on the residual <code>sigma</code> and the group standard deviation. Call <code>brms::prior_summary(ex_3_2)</code> to retrieve the prior table and save the returned data frame to <code>ex_3_4</code>.</p>
<div class="exercise-expected">
<p><strong>Expected result:</strong></p>
<pre><code>#> prior class coef group resp dpar nlpar lb ub source
#> (flat) b default
#> (flat) b wt (vectorized)
#> student_t(3, 19.2, 5.4) Intercept default
#> student_t(3, 0, 5.4) sd default
#> student_t(3, 0, 5.4) sd cyl 0 (vectorized)
#> student_t(3, 0, 5.4) sd Intercept cyl 0 (vectorized)
#> student_t(3, 0, 5.4) sigma 0 default</code></pre>
</div>
<p><strong>Difficulty:</strong> Intermediate</p>
<div class="exercise-hints" hidden><p>Audit the priors brms actually assigned, including the defaults it silently filled in for the variance terms.</p><p>Call <code>brms::prior_summary()</code> on the fitted hierarchical model <code>ex_3_2</code>.</p></div>