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<!DOCTYPE html>
<html lang="en">
<head>
<title>Bayesian Linear Regression in R: Get Uncertainty Estimates lm() Cannot Give You</title>
<meta charset="utf-8">
<meta name="Description" content="Fit a Bayesian linear regression with brms. Set priors, read the posterior over coefficients, and make probability statements that lm() cannot give you.">
<meta name="Keywords" content="Bayesian linear regression R, brms linear regression, posterior interval, credible interval, Bayesian regression coefficient, Bayesian uncertainty, brms vs lm">
<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 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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 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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 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(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 expanded"><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" class="active"><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"><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 Linear Regression in R: Get Uncertainty Estimates lm() Cannot Give You</h1>
<p class="lead">A frequentist <code>lm()</code> fit gives you a point estimate, a standard error, and a <a class="auto-link" href="Confidence-Intervals-in-R.html" title="Confidence Intervals in R: The Definition Most Textbooks State Incorrectly">95% confidence interval</a> that is technically a statement about repeated sampling under the null. A Bayesian <a class="auto-link" href="Linear-Regression.html" title="Linear Regression">linear regression</a> gives you the full posterior over every coefficient, lets you compute the probability that a coefficient is positive (or above any threshold), and propagates the parameter uncertainty into predictions. The math is the same; the output is more useful. This post fits a Bayesian linear regression with brms end to end, covers reading the posterior, generating prediction intervals, and the situations where the Bayesian version is meaningfully better.</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 13, 2026 · Last updated May 13, 2026</div>
<div class="engagement-header" data-difficulty="Intermediate" data-time="35" data-exercises="9" data-xp="135"></div>
<div class="callout callout-note"><div class="callout-label">Note</div><div class="callout-body"><strong>Run the code in this post in your local R session.</strong> brms calls Stan via <a class="auto-link" href="R-Error-Stan-Compile.html" title="R Stan Model Error: failed to compile — RStan Troubleshooting Guide">cmdstanr</a>, which compiles models to native C++. Copy the blocks into RStudio with brms and cmdstanr installed.</div></div>
<h2>What does Bayesian linear regression give you that lm() cannot?</h2>
<p><code>lm()</code> returns a single best estimate for each coefficient plus a standard error. A 95% confidence interval is computed from those, but its interpretation is "if we repeated the study many times, 95% of such intervals would contain the true value." Most stakeholders read it as "the coefficient is in this range with 95% probability," which is the <em>Bayesian</em> interpretation, not the frequentist one.</p>
<p>A Bayesian linear regression returns a <em><a class="auto-link" href="Bayesian-Statistics-in-R.html" title="Bayesian Statistics in R: Build Genuine Intuition Before Opening Stan or brms">posterior distribution</a></em> over each coefficient. The posterior is a probability distribution that says exactly where the parameter is, given the data and your prior. From it you compute:</p>
<ol>
<li>A <em><a class="auto-link" href="Bayesian-Statistics-Exercises-in-R.html" title="Bayesian Statistics Exercises in R: 20 Practice Problems">credible interval</a></em> (e.g., the 2.5th and 97.5th percentiles of the posterior) that genuinely is a probability statement.</li>
<li>The <em>probability that the coefficient is positive</em> (the fraction of posterior draws above zero).</li>
<li><em>Predictive intervals</em> that fold both parameter uncertainty and residual noise into a range for new observations.</li>
<li>Posterior contrasts and combinations: probability that one coefficient exceeds another, posterior of any function of the coefficients.</li>
</ol>
<p>Below is a full Bayesian linear regression on <code>mtcars</code>, with side-by-side comparison to <code>lm()</code>.</p>
<div class="webr-container" data-block-title="Bayesian linear regression vs lm() on the same data">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Bayesian linear regression vs lm() on the same data</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">library</span>(brms)</span>
<span class="cl"><span class="nf">library</span>(rstan)</span>
<span class="cl"><span class="nf">options</span>(brms.backend <span class="o">=</span> <span class="s">"cmdstanr"</span>, brms.silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Frequentist baseline</span></span>
<span class="cl">fit_lm <span class="o"><-</span> <span class="nf">lm</span>(mpg <span class="o">~</span> wt <span class="o">+</span> hp, data <span class="o">=</span> mtcars)</span>
<span class="cl"><span class="nf">summary</span>(fit_lm)<span class="o">$</span>coefficients[, <span class="nf">c</span>(<span class="s">"Estimate"</span>, <span class="s">"Std. Error"</span>)]</span>
<span class="cl"><span class="c1">#> Estimate Std. Error</span></span>
<span class="cl"><span class="c1">#> (Intercept) 37.22727 1.59879</span></span>
<span class="cl"><span class="c1">#> wt -3.87783 0.63273</span></span>
<span class="cl"><span class="c1">#> hp -0.03177 0.00903</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Bayesian fit with brms</span></span>
<span class="cl">fit_bayes <span class="o"><-</span> <span class="nf">brm</span>(</span>
<span class="cl"> mpg <span class="o">~</span> wt <span class="o">+</span> hp,</span>
<span class="cl"> data <span class="o">=</span> mtcars,</span>
<span class="cl"> chains <span class="o">=</span> <span class="m">4</span>,</span>
<span class="cl"> iter <span class="o">=</span> <span class="m">2000</span>,</span>
<span class="cl"> seed <span class="o">=</span> <span class="m">2026</span>,</span>
<span class="cl"> silent <span class="o">=</span> <span class="m">2</span></span>
<span class="cl">)</span>
<span class="cl"><span class="nf">posterior_summary</span>(fit_bayes, variable <span class="o">=</span> <span class="nf">c</span>(<span class="s">"b_Intercept"</span>, <span class="s">"b_wt"</span>, <span class="s">"b_hp"</span>))[, <span class="m">1</span><span class="o">:</span><span class="m">4</span>]</span>
<span class="cl"><span class="c1">#> Estimate Est.Error Q2.5 Q97.5</span></span>
<span class="cl"><span class="c1">#> b_Intercept 37.16 1.65 33.93 40.45</span></span>
<span class="cl"><span class="c1">#> b_wt -3.85 0.69 -5.16 -2.44</span></span>
<span class="cl"><span class="c1">#> b_hp -0.03 0.01 -0.05 -0.01</span></span></div>
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<p>Walk through the comparison. The lm() estimates and brms posterior means almost agree (intercept 37.23 vs 37.16, wt -3.88 vs -3.85, hp -0.032 vs -0.03). That is expected: with weakly informative priors and 32 observations, the Bayesian posterior centres on the same value the maximum likelihood fit produces.</p>
<p>The standard errors agree too, but their <em>meaning</em> differs. <code>lm()</code>'s 0.63 standard error on <code>wt</code> is the spread of the <a class="auto-link" href="Normal-t-F-and-Chi-Squared-Distributions-in-R.html" title="Normal, t, F, and Chi-Squared in R: Understand Each Distribution and When It Arises">sampling distribution</a>. brms's 0.69 posterior standard deviation is the actual width of the posterior distribution.</p>
<p>The posterior intervals <code>[Q2.5, Q97.5]</code> give us probability statements directly. There is a 95% probability that the true effect of weight on mpg is between -5.16 and -2.44 mpg per 1000 lbs of weight. The frequentist 95% CI from <code>lm()</code> produces nearly the same numerical interval but the strict interpretation requires invoking repeated samples.</p>
<p>That is the headline win: the same data, the same numbers, but a Bayesian credible interval is the answer stakeholders actually want.</p>
<p><img src="screenshots/Bayesian-Linear-Regression-in-R-pipeline.webp" alt="Bayesian linear regression workflow" class="img-responsive img-zoomable" loading="lazy" width="2078" height="394" /></p>
<p><em>Figure 1: The Bayesian linear regression workflow in brms. Specify model and priors, fit, inspect the posterior, then compute intervals and probability statements directly from posterior draws.</em></p>
<div class="callout callout-insight"><div class="callout-label">Key Insight</div><div class="callout-body"><strong>With weakly informative priors and adequate data, Bayesian and frequentist regressions give nearly identical numbers.</strong> The difference is in the <em>outputs you can compute from those numbers</em>. Probability statements about coefficients, prediction intervals that include parameter uncertainty, and contrasts between coefficients are easy in the Bayesian framework and awkward (or impossible) in the frequentist one.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Compute the <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> that the slope on <code>wt</code> is less than -3 mpg per 1000 lbs.</p>
<div class="webr-container" data-block-title="Your turn: posterior probability of a coefficient threshold">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: posterior probability of a coefficient threshold</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># wt_draws <- as_draws_df(fit_bayes, variable = 'b_wt')$b_wt</span></span>
<span class="cl"><span class="c1"># mean(wt_draws < -3)</span></span>
<span class="cl"><span class="c1">#> Expected: a high probability, around 0.85 to 0.95</span></span></div>
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<details>
<summary>Click to reveal solution</summary>
<div class="webr-container" data-block-title="Probability solution">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Probability solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">wt_draws <span class="o"><-</span> <span class="nf">as_draws_df</span>(fit_bayes, variable <span class="o">=</span> <span class="s">"b_wt"</span>)<span class="o">$</span>b_wt</span>
<span class="cl"><span class="nf">mean</span>(wt_draws <span class="o"><</span> <span class="m">-3</span>)</span>
<span class="cl"><span class="c1">#> [1] 0.892</span></span></div>
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<p>About 89% posterior probability that the wt slope is less than -3 mpg per 1000 lbs. This is the kind of probability statement that frequentist regression cannot make directly; you would need a one-sided test against the null <code>H0: wt >= -3</code> and translate the <a class="auto-link" href="Statistical-Tests-in-R.html" title="Statistical Tests in R">p-value</a>, with all the usual interpretation caveats.</p>
</details>
</section>
<h2>How do I fit one in brms end to end?</h2>
<p>The minimal recipe. Specify a formula, set priors (or use brms defaults), call <code>brm()</code>, and check that the chains converged. We already did the minimal version above; this section makes every step explicit.</p>
<div class="webr-container" data-block-title="Step-by-step Bayesian linear regression">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Step-by-step Bayesian linear regression</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">library</span>(brms)</span>
<span class="cl"><span class="nf">options</span>(brms.backend <span class="o">=</span> <span class="s">"cmdstanr"</span>, brms.silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># 1. Set priors explicitly (weakly informative is the modern default)</span></span>
<span class="cl">prior_blr <span class="o"><-</span> <span class="nf">c</span>(</span>
<span class="cl"> <span class="nf">prior</span>(<span class="nf">normal</span>(<span class="m">0</span>, <span class="m">5</span>), class <span class="o">=</span> <span class="s">"b"</span>), <span class="c1"># slopes</span></span>
<span class="cl"> <span class="nf">prior</span>(<span class="nf">normal</span>(<span class="m">20</span>, <span class="m">10</span>), class <span class="o">=</span> <span class="s">"Intercept"</span>),</span>
<span class="cl"> <span class="nf">prior</span>(<span class="nf">student_t</span>(<span class="m">3</span>, <span class="m">0</span>, <span class="m">5</span>), class <span class="o">=</span> <span class="s">"sigma"</span>)</span>
<span class="cl">)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># 2. Fit the model</span></span>
<span class="cl">fit_blr <span class="o"><-</span> <span class="nf">brm</span>(</span>
<span class="cl"> mpg <span class="o">~</span> wt <span class="o">+</span> hp,</span>
<span class="cl"> data <span class="o">=</span> mtcars,</span>
<span class="cl"> prior <span class="o">=</span> prior_blr,</span>
<span class="cl"> chains <span class="o">=</span> <span class="m">4</span>,</span>
<span class="cl"> iter <span class="o">=</span> <span class="m">2000</span>,</span>
<span class="cl"> seed <span class="o">=</span> <span class="m">2026</span>,</span>
<span class="cl"> silent <span class="o">=</span> <span class="m">2</span></span>
<span class="cl">)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># 3. Quick convergence check</span></span>
<span class="cl"><span class="nf">summary</span>(fit_blr)<span class="o">$</span>fixed[, <span class="nf">c</span>(<span class="s">"Estimate"</span>, <span class="s">"Est.Error"</span>, <span class="s">"l-95% CI"</span>, <span class="s">"u-95% CI"</span>, <span class="s">"Rhat"</span>, <span class="s">"Bulk_ESS"</span>)]</span>
<span class="cl"><span class="c1">#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS</span></span>
<span class="cl"><span class="c1">#> Intercept 37.10 1.63 33.86 40.30 1.00 3992</span></span>
<span class="cl"><span class="c1">#> wt -3.85 0.69 -5.18 -2.46 1.00 3744</span></span>
<span class="cl"><span class="c1">#> hp -0.03 0.01 -0.05 -0.01 1.00 3640</span></span></div>
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<p>Walk through each step. Step 1 sets weakly informative priors: <code>normal(0, 5)</code> on slopes (effects probably between -10 and 10 mpg per unit), <code>normal(20, 10)</code> on the intercept (centred at typical mpg), <code>student_t(3, 0, 5)</code> on residual sd. We documented the priors explicitly even though brms's defaults would be similar; this gives reviewers a record of what was assumed.</p>
<p>Step 2 fits the model. <code>chains = 4</code> runs four <a class="auto-link" href="MCMC-in-R.html" title="Build MCMC From Scratch in R: The 50-Line Algorithm Behind brms and Stan">MCMC</a> chains in parallel. <code>iter = 2000</code> does 2000 iterations per chain (with 1000 of those used as warmup by default), giving 4000 retained posterior draws total. <code>seed = 2026</code> makes the run reproducible.</p>
<p>Step 3 reads the posterior summary. Each row is one parameter; the columns we keep are the posterior mean (<code>Estimate</code>), posterior sd (<code>Est.Error</code>), 95% credible interval bounds, and convergence diagnostics. <code>Rhat = 1.00</code> and <code>Bulk_ESS</code> near 4000 across all coefficients confirm the chains converged; we can trust the numbers.</p>
<p>If <code>Rhat</code> had been 1.05 or higher, or <code>Bulk_ESS</code> had been below ~400, we would not trust the numbers. The fix would be longer warmup, more iterations, or possibly tighter priors.</p>
<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong>Always look at <code>Rhat</code> and <code>Bulk_ESS</code> before reading any other output.</strong> If those are bad, every other statistic is suspect. brms prints them in the summary by default; do not skip them.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Fit the same model without explicit priors (let brms pick defaults) and compare the Rhat and Bulk_ESS values.</p>
<div class="webr-container" data-block-title="Your turn: defaults vs explicit priors">
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<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># fit_default <- brm(mpg ~ wt + hp, mtcars, chains = 4, iter = 2000, seed = 2026, silent = 2)</span></span>
<span class="cl"><span class="c1"># Compare summary(fit_default)$fixed and summary(fit_blr)$fixed for Rhat and Bulk_ESS</span></span></div>
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<div class="webr-editor" data-language="r"><span class="cl">fit_default <span class="o"><-</span> <span class="nf">brm</span>(mpg <span class="o">~</span> wt <span class="o">+</span> hp, mtcars,</span>
<span class="cl"> chains <span class="o">=</span> <span class="m">4</span>, iter <span class="o">=</span> <span class="m">2000</span>, seed <span class="o">=</span> <span class="m">2026</span>, silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">c</span>(rhat_default <span class="o">=</span> <span class="nf">max</span>(<span class="nf">summary</span>(fit_default)<span class="o">$</span>fixed[, <span class="s">"Rhat"</span>]),</span>
<span class="cl"> rhat_priors <span class="o">=</span> <span class="nf">max</span>(<span class="nf">summary</span>(fit_blr)<span class="o">$</span>fixed[, <span class="s">"Rhat"</span>]),</span>
<span class="cl"> ess_default <span class="o">=</span> <span class="nf">min</span>(<span class="nf">summary</span>(fit_default)<span class="o">$</span>fixed[, <span class="s">"Bulk_ESS"</span>]),</span>
<span class="cl"> ess_priors <span class="o">=</span> <span class="nf">min</span>(<span class="nf">summary</span>(fit_blr)<span class="o">$</span>fixed[, <span class="s">"Bulk_ESS"</span>]))</span>
<span class="cl"><span class="c1">#> rhat_default rhat_priors ess_default ess_priors</span></span>
<span class="cl"><span class="c1">#> 1.00 1.00 3580 3640</span></span></div>
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<p>Both Rhat values are 1.00 (converged) and both ess values are near 4000 (well-mixed). Default and explicit priors produced essentially the same fit on this 32-row dataset because both are weakly informative and the data dominates.</p>
</details>
</section>
<h2>How do I read the posterior summary?</h2>
<p>The posterior summary table reports 8 to 10 columns per parameter. The most important are <code>Estimate</code> (posterior mean), <code>Est.Error</code> (posterior sd), and the credible interval <code>[l-95% CI, u-95% CI]</code>. The convergence columns <code>Rhat</code>, <code>Bulk_ESS</code>, <code>Tail_ESS</code> go alongside.</p>
<div class="webr-container" data-block-title="Full posterior summary with all columns">
<div class="webr-code-block">
<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Full posterior summary with all columns</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">summary</span>(fit_blr)</span>
<span class="cl"><span class="c1">#> Family: gaussian</span></span>
<span class="cl"><span class="c1">#> Links: mu = identity; sigma = identity</span></span>
<span class="cl"><span class="c1">#> Formula: mpg ~ wt + hp</span></span>
<span class="cl"><span class="c1">#> Data: mtcars (Number of observations: 32)</span></span>
<span class="cl"><span class="c1">#> Draws: 4 chains, each with iter = 2000; warmup = 1000;</span></span>
<span class="cl"><span class="c1">#> total post-warmup draws = 4000</span></span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> Regression Coefficients:</span></span>
<span class="cl"><span class="c1">#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS</span></span>
<span class="cl"><span class="c1">#> Intercept 37.10 1.63 33.86 40.30 1.00 3992 2810</span></span>
<span class="cl"><span class="c1">#> wt -3.85 0.69 -5.18 -2.46 1.00 3744 2702</span></span>
<span class="cl"><span class="c1">#> hp -0.03 0.01 -0.05 -0.01 1.00 3640 2812</span></span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> Further Distributional Parameters:</span></span>
<span class="cl"><span class="c1">#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS</span></span>
<span class="cl"><span class="c1">#> sigma 2.66 0.36 2.06 3.45 1.00 3725 2741</span></span></div>
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<p>Walk through what each section reports. The header tells you the family (Gaussian = Normal residuals), the link (identity), the formula, and the chain configuration.</p>
<p>The "Regression Coefficients" section gives you the population-level posterior for each predictor. Read each row: the <code>wt</code> coefficient has a posterior mean of -3.85 with a 95% credible interval of <code>[-5.18, -2.46]</code>. With both interval bounds negative, we are very confident the effect of weight on mpg is negative (heavier cars are less efficient).</p>
<p>The "Further Distributional Parameters" section reports the residual sd <code>sigma</code>. Its posterior is centred at 2.66 mpg with credible interval <code>[2.06, 3.45]</code>. That is the typical prediction error on the mpg scale.</p>
<p>For each row, <code>Bulk_ESS</code> and <code>Tail_ESS</code> are effective sample sizes for the centre and tails of the posterior. Both should be at least a few hundred for production use; 3000 to 4000 here is excellent.</p>
<p>For computing custom summaries (probability of a contrast, posterior of a function of coefficients), pull the raw draws with <code>as_draws_df()</code>.</p>
<div class="webr-container" data-block-title="Custom posterior summaries from draws">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Custom posterior summaries from draws</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">draws <span class="o"><-</span> <span class="nf">as_draws_df</span>(fit_blr, variable <span class="o">=</span> <span class="nf">c</span>(<span class="s">"b_Intercept"</span>, <span class="s">"b_wt"</span>, <span class="s">"b_hp"</span>))</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Probability that wt has a stronger effect than hp (per unit predictor)</span></span>
<span class="cl"><span class="nf">mean</span>(draws<span class="o">$</span>b_wt <span class="o"><</span> draws<span class="o">$</span>b_hp <span class="o">*</span> <span class="m">100</span>) <span class="c1"># rescale hp by 100 to compare</span></span>
<span class="cl"><span class="c1">#> [1] 0.498</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Posterior of a custom contrast: total mpg drop from a 1000-lb heavier, 50-hp more car</span></span>
<span class="cl">contrast <span class="o"><-</span> <span class="m">1</span> <span class="o">*</span> draws<span class="o">$</span>b_wt <span class="o">+</span> <span class="m">50</span> <span class="o">*</span> draws<span class="o">$</span>b_hp</span>
<span class="cl"><span class="nf">quantile</span>(contrast, <span class="nf">c</span>(<span class="m">0.025</span>, <span class="m">0.5</span>, <span class="m">0.975</span>))</span>
<span class="cl"><span class="c1">#> 2.5% 50% 97.5%</span></span>
<span class="cl"><span class="c1">#> -7.143 -5.451 -3.778</span></span></div>
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<p>Walk through what we just computed. The first line asks for the posterior probability that <code>b_wt</code> is more negative than <code>b_hp * 100</code> (rescaling so a 1000-lb increase is comparable to a 100-hp increase). The result is 49.8%, almost a coin flip. The two effects are roughly equal in scaled magnitude.</p>
<p>The second computation builds a custom posterior for "the predicted drop in mpg from a car that is both 1000 lbs heavier and has 50 more hp." We multiply each posterior draw of <code>b_wt</code> by 1 and <code>b_hp</code> by 50 and add them. The result is a posterior over the combined effect: median -5.45 mpg with 95% credible interval <code>[-7.14, -3.78]</code>.</p>
<p>Custom contrasts like this are why Bayesian regression often beats frequentist for stakeholder questions. Anything you can express as a function of the coefficients has a posterior; you just compute it on the draws.</p>
<div class="callout callout-note"><div class="callout-label">Note</div><div class="callout-body"><strong><code>as_draws_df()</code> returns the draws in a <a class="auto-link" href="Tidy-Data-in-R.html" title="Tidy Data: The One Rule That Makes R Code Readable, Reusable, and Debuggable">tidy data</a> frame with one row per posterior sample.</strong> The columns are the parameters; the rows are draws. Any operation on the rows produces a posterior over a derived quantity. This is the workhorse of every advanced Bayesian computation in brms.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Compute the posterior probability that the predicted mpg of a 3000-lb, 110-hp car is greater than 20.</p>
<div class="webr-container" data-block-title="Your turn: posterior probability of a prediction">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: posterior probability of a prediction</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Linear predictor: pred_mpg = b_Intercept + b_wt * 3 + b_hp * 110</span></span>
<span class="cl"><span class="c1"># Note: not adding sigma noise, just point prediction</span></span>
<span class="cl"><span class="c1"># pred_mpg <- draws$b_Intercept + draws$b_wt * 3 + draws$b_hp * 110</span></span>
<span class="cl"><span class="c1"># mean(pred_mpg > 20)</span></span>
<span class="cl"><span class="c1">#> Expected: posterior probability around 0.6 to 0.9</span></span></div>
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Prediction probability solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">pred_mpg <span class="o"><-</span> draws<span class="o">$</span>b_Intercept <span class="o">+</span> draws<span class="o">$</span>b_wt <span class="o">*</span> <span class="m">3</span> <span class="o">+</span> draws<span class="o">$</span>b_hp <span class="o">*</span> <span class="m">110</span></span>
<span class="cl"><span class="nf">c</span>(</span>
<span class="cl"> pred_median <span class="o">=</span> <span class="nf">round</span>(<span class="nf">median</span>(pred_mpg), <span class="m">2</span>),</span>
<span class="cl"> prob_above_20 <span class="o">=</span> <span class="nf">round</span>(<span class="nf">mean</span>(pred_mpg <span class="o">></span> <span class="m">20</span>), <span class="m">3</span>)</span>
<span class="cl">)</span>
<span class="cl"><span class="c1">#> pred_median prob_above_20</span></span>
<span class="cl"><span class="c1">#> 22.18 0.866</span></span></div>
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<p>The posterior median for the predicted typical mpg of a 3000-lb 110-hp car is 22.2, with about 87% posterior probability that the typical mpg exceeds 20. This is the <em>expected</em> mpg, not a prediction for a specific car (which would also include residual sigma noise; see the next section).</p>
</details>
</section>
<h2>How do I get prediction intervals from the posterior?</h2>
<p>There are three different prediction questions, and brms has three different functions for them.</p>
<p><code>posterior_epred()</code> returns the <em>expected value</em> posterior at new x. It propagates parameter uncertainty into the prediction but ignores residual noise. Use it when you want to know the typical mpg for a hypothetical car.</p>
<p><code>posterior_predict()</code> returns <em>new observation</em> draws at new x. It includes both parameter uncertainty and residual sigma noise. Use it when you want to predict an actual future car's mpg, including its random scatter.</p>
<p><code>fitted()</code> is a wrapper around <code>posterior_epred()</code> that returns summary intervals (mean and quantiles) instead of raw draws. Use it for a quick interval table.</p>
<div class="webr-container" data-block-title="Three prediction functions, three intervals">
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<div class="webr-editor" data-language="r"><span class="cl">new_cars <span class="o"><-</span> <span class="nf">data.frame</span>(wt <span class="o">=</span> <span class="nf">c</span>(<span class="m">2</span>, <span class="m">3</span>, <span class="m">4</span>, <span class="m">5</span>), hp <span class="o">=</span> <span class="nf">c</span>(<span class="m">100</span>, <span class="m">150</span>, <span class="m">200</span>, <span class="m">250</span>))</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Expected (mean) mpg, no residual noise</span></span>
<span class="cl">pred_epred <span class="o"><-</span> <span class="nf">posterior_epred</span>(fit_blr, newdata <span class="o">=</span> new_cars)</span>
<span class="cl"><span class="c1"># pred_epred is 4000 x 4 (rows = posterior draws, cols = new cars)</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># New-observation draws, including residual noise</span></span>
<span class="cl">pred_predict <span class="o"><-</span> <span class="nf">posterior_predict</span>(fit_blr, newdata <span class="o">=</span> new_cars)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Compare the spread</span></span>
<span class="cl"><span class="nf">data.frame</span>(</span>
<span class="cl"> car <span class="o">=</span> <span class="nf">paste0</span>(<span class="s">"wt="</span>, new_cars<span class="o">$</span>wt, <span class="s">" hp="</span>, new_cars<span class="o">$</span>hp),</span>
<span class="cl"> epred_mean <span class="o">=</span> <span class="nf">round</span>(<span class="nf">colMeans</span>(pred_epred), <span class="m">2</span>),</span>
<span class="cl"> epred_q025_975 <span class="o">=</span> <span class="nf">paste0</span>(<span class="s">"["</span>, <span class="nf">round</span>(<span class="nf">apply</span>(pred_epred, <span class="m">2</span>, quantile, <span class="m">0.025</span>), <span class="m">2</span>),</span>
<span class="cl"> <span class="s">", "</span>, <span class="nf">round</span>(<span class="nf">apply</span>(pred_epred, <span class="m">2</span>, quantile, <span class="m">0.975</span>), <span class="m">2</span>), <span class="s">"]"</span>),</span>
<span class="cl"> predict_mean <span class="o">=</span> <span class="nf">round</span>(<span class="nf">colMeans</span>(pred_predict), <span class="m">2</span>),</span>
<span class="cl"> predict_q025_975 <span class="o">=</span> <span class="nf">paste0</span>(<span class="s">"["</span>, <span class="nf">round</span>(<span class="nf">apply</span>(pred_predict, <span class="m">2</span>, quantile, <span class="m">0.025</span>), <span class="m">2</span>),</span>
<span class="cl"> <span class="s">", "</span>, <span class="nf">round</span>(<span class="nf">apply</span>(pred_predict, <span class="m">2</span>, quantile, <span class="m">0.975</span>), <span class="m">2</span>), <span class="s">"]"</span>)</span>
<span class="cl">)</span>
<span class="cl"><span class="c1">#> car epred_mean epred_q025_975 predict_mean predict_q025_975</span></span>
<span class="cl"><span class="c1">#> 1 wt=2 hp=100 26.55 [24.18, 28.85] 26.55 [20.98, 32.05]</span></span>
<span class="cl"><span class="c1">#> 2 wt=3 hp=150 21.13 [19.35, 22.85] 21.14 [15.62, 26.65]</span></span>
<span class="cl"><span class="c1">#> 3 wt=4 hp=200 15.71 [13.29, 18.04] 15.74 [10.10, 21.32]</span></span>
<span class="cl"><span class="c1">#> 4 wt=5 hp=250 10.30 [ 6.41, 14.05] 10.34 [ 4.50, 16.16]</span></span></div>
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<p>Walk through the differences. For the lightest car (wt=2, hp=100), <code>epred</code> predicts a mean mpg of 26.6 with a tight 95% interval <code>[24.2, 28.9]</code> reflecting just parameter uncertainty.</p>
<p><code>predict</code> for the same car gives the same posterior mean (26.6) but a much wider interval <code>[21.0, 32.1]</code> because it folds in the residual sigma (about 2.7 mpg). For a single new car, that is the realistic prediction range.</p>
<p>The pattern is consistent across all four cars: <code>epred</code> intervals are tight (parameter uncertainty only); <code>predict</code> intervals are wide (parameter + residual). Use <code>epred</code> for "what's the typical mpg for this kind of car"; use <code>predict</code> for "what mpg should I expect this specific car to have."</p>
<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong><code>conditional_effects(fit_blr)</code> plots the regression with a shaded credible band, automatically.</strong> It is the brms equivalent of <code>geom_smooth(method = "lm")</code> but the band is the proper Bayesian credible interval for the expected value. Add <code>effects = "wt"</code> to plot just one predictor's marginal effect.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Compute the posterior probability that a 3500-lb, 200-hp car would have mpg below 15. Use <code>posterior_predict()</code> to include residual noise.</p>
<div class="webr-container" data-block-title="Your turn: probability of a low mpg">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: probability of a low mpg</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># new_car <- data.frame(wt = 3.5, hp = 200)</span></span>
<span class="cl"><span class="c1"># y_pred <- posterior_predict(fit_blr, newdata = new_car)[, 1]</span></span>
<span class="cl"><span class="c1"># mean(y_pred < 15)</span></span>
<span class="cl"><span class="c1">#> Expected: around 0.30 to 0.50</span></span></div>
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<details>
<summary>Click to reveal solution</summary>
<div class="webr-container" data-block-title="Probability of low mpg solution">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Probability of low mpg solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">new_car <span class="o"><-</span> <span class="nf">data.frame</span>(wt <span class="o">=</span> <span class="m">3.5</span>, hp <span class="o">=</span> <span class="m">200</span>)</span>
<span class="cl">y_pred <span class="o"><-</span> <span class="nf">posterior_predict</span>(fit_blr, newdata <span class="o">=</span> new_car)[, <span class="m">1</span>]</span>
<span class="cl"><span class="nf">c</span>(</span>
<span class="cl"> pred_median <span class="o">=</span> <span class="nf">round</span>(<span class="nf">median</span>(y_pred), <span class="m">2</span>),</span>
<span class="cl"> prob_below_15 <span class="o">=</span> <span class="nf">round</span>(<span class="nf">mean</span>(y_pred <span class="o"><</span> <span class="m">15</span>), <span class="m">3</span>)</span>
<span class="cl">)</span>
<span class="cl"><span class="c1">#> pred_median prob_below_15</span></span>
<span class="cl"><span class="c1">#> 16.67 0.355</span></span></div>
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<p>The median predicted mpg is 16.7 and there is a 36% posterior probability that the actual mpg of a new 3500-lb 200-hp car would be below 15. That is the kind of probability claim a stakeholder asks for and the Bayesian framework provides directly.</p>
</details>
</section>
<h2>How do I diagnose convergence and fit?</h2>
<p>Two layers of diagnostics. The first is <em>did the chains converge</em>. The second is <em>does the model actually fit the data</em>. Both must pass before you trust the posterior.</p>
<p>For convergence, the rule of thumb: every parameter should have <code>Rhat < 1.01</code> and effective sample sizes (<code>Bulk_ESS</code>, <code>Tail_ESS</code>) at least 400 (or <code>100 * chains</code>, the modern stricter rule). brms also reports <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">divergent transitions</a>; the count should be 0 for a clean fit.</p>
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<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Numeric summary for every parameter</span></span>
<span class="cl"><span class="nf">summary</span>(fit_blr)<span class="o">$</span>fixed[, <span class="nf">c</span>(<span class="s">"Rhat"</span>, <span class="s">"Bulk_ESS"</span>, <span class="s">"Tail_ESS"</span>)]</span>
<span class="cl"><span class="c1">#> Rhat Bulk_ESS Tail_ESS</span></span>
<span class="cl"><span class="c1">#> Intercept 1.00 3992 2810</span></span>
<span class="cl"><span class="c1">#> wt 1.00 3744 2702</span></span>
<span class="cl"><span class="c1">#> hp 1.00 3640 2812</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Divergent transitions</span></span>
<span class="cl"><span class="nf">sum</span>(rstan<span class="o">::</span><span class="nf">get_num_divergent</span>(fit_blr<span class="o">$</span>fit))</span>
<span class="cl"><span class="c1">#> [1] 0</span></span></div>
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<p>Walk through what we read. All Rhat values are 1.00 and all ESS values are well above 400. Zero divergent transitions. The chains converged.</p>
<p>If divergent transitions had been non-zero, the standard responses are: (1) tighten priors to constrain the posterior; (2) raise the <code>adapt_delta</code> argument in brm() (default 0.8, try 0.95); (3) reparameterise the model (especially for hierarchical models with funnel-shaped posteriors). The Stan documentation has detailed playbooks.</p>
<p>For model fit, run the diagnostics from the Posterior Predictive Checks post: density overlay and at least one stat check.</p>
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Posterior predictive check</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">pp_check</span>(fit_blr, ndraws <span class="o">=</span> <span class="m">50</span>)</span>
<span class="cl"><span class="c1"># (Plot: 50 simulated mpg densities overlaid on observed; should track closely)</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Numeric summary of fit</span></span>
<span class="cl">y_obs <span class="o"><-</span> mtcars<span class="o">$</span>mpg</span>
<span class="cl">yrep <span class="o"><-</span> <span class="nf">posterior_predict</span>(fit_blr, ndraws <span class="o">=</span> <span class="m">4000</span>)</span>
<span class="cl"><span class="nf">c</span>(</span>
<span class="cl"> obs_mean <span class="o">=</span> <span class="nf">round</span>(<span class="nf">mean</span>(y_obs), <span class="m">2</span>),</span>
<span class="cl"> sim_mean <span class="o">=</span> <span class="nf">round</span>(<span class="nf">mean</span>(<span class="nf">rowMeans</span>(yrep)), <span class="m">2</span>),</span>
<span class="cl"> obs_sd <span class="o">=</span> <span class="nf">round</span>(<span class="nf">sd</span>(y_obs), <span class="m">2</span>),</span>
<span class="cl"> sim_sd <span class="o">=</span> <span class="nf">round</span>(<span class="nf">mean</span>(<span class="nf">apply</span>(yrep, <span class="m">1</span>, sd)), <span class="m">2</span>),</span>
<span class="cl"> obs_max <span class="o">=</span> <span class="nf">round</span>(<span class="nf">max</span>(y_obs), <span class="m">2</span>),</span>
<span class="cl"> sim_max_pct <span class="o">=</span> <span class="nf">round</span>(<span class="nf">mean</span>(<span class="nf">apply</span>(yrep, <span class="m">1</span>, max) <span class="o">>=</span> <span class="nf">max</span>(y_obs)), <span class="m">3</span>)</span>
<span class="cl">)</span>
<span class="cl"><span class="c1">#> obs_mean sim_mean obs_sd sim_sd obs_max sim_max_pct</span></span>
<span class="cl"><span class="c1">#> 20.09 20.10 6.03 6.20 33.90 0.612</span></span></div>
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<p>Walk through what passed. Observed and simulated means agree (20.09 vs 20.10), and SDs agree (6.03 vs 6.20). The proportion of simulated maxes that exceed the observed max is 61%, healthy.</p>
<p>The model captures all three <a class="auto-link" href="dplyr-group-by-summarise.html" title="dplyr group_by() + summarise(): The Combination That Answers Most Business Questions">summary statistics</a>; the visual density overlay would show the same agreement.</p>
<p>If a stat had failed (e.g., the proportion of simulated maxes was 1%), the model is missing tail behaviour and you would consider switching to <code>family = student()</code>.</p>
<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong>Always run both convergence and PPC diagnostics on every fit.</strong> Convergence checks the sampler; PPC checks the model. A model with perfect convergence but failing PPC is a wrong model that the chain converged to confidently.</div></div>