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<!DOCTYPE html>
<html lang="en">
<head>
<title>Compare Bayesian Models in R: Why LOO and WAIC Beat AIC for Bayesian Fits</title>
<meta charset="utf-8">
<meta name="Description" content="LOO-CV estimates out-of-sample predictive accuracy from a single fit. Use loo() and waic() in R to compare Bayesian models with honest standard errors.">
<meta name="Keywords" content="compare Bayesian models R, LOO-CV R, WAIC R, loo package R, brms loo_compare, elpd_loo, Pareto k diagnostic, Bayesian model comparison, AIC vs LOO">
<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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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" class="active"><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"><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>Compare Bayesian Models in R: Why LOO and WAIC Beat AIC for Bayesian Fits</h1>
<p class="lead">You fit two or three Bayesian models on the same data. Which one do you report? For frequentist fits, the standard answer is <a class="auto-link" href="Model-Selection-in-R.html" title="Model Selection in R">AIC</a>, but AIC uses a point estimate of the parameters and ignores the posterior. The Bayesian replacement is leave-one-out cross-validation (LOO-CV), implemented in R by the <code>loo</code> package. It estimates how well your model predicts new data without actually refitting, gives a proper standard error on the comparison, and warns you when its own approximation is unreliable. This post walks through computing LOO and WAIC for brms models, reading the comparison output, and recognising the failure modes.</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 10, 2026 · Last updated May 10, 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> The brms calls invoke 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, cmdstanr, and the loo package installed.</div></div>
<h2>Why does AIC fail for Bayesian models?</h2>
<p>AIC (Akaike Information Criterion) is <code>-2 * log-likelihood + 2 * k</code>, where k is the number of parameters. The log-likelihood is computed at the maximum likelihood estimate (MLE), a single point in parameter space. AIC penalises adding parameters by <code>2 * k</code>, which is correct under a specific asymptotic argument about the MLE-fit model on new data.</p>
<p>The trouble is that a Bayesian fit does not have an MLE. It has a <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> over parameters. AIC has no clean way to use that posterior, so applied to a Bayesian fit, you would either reduce the posterior to a point (losing all uncertainty information) or use the posterior mean of the log-likelihood (which has no well-defined penalty).</p>
<p>The Bayesian replacement is to estimate the <em>expected log predictive density</em> (elpd) of the model on new data using the full posterior. There are two practical methods.</p>
<p>The first is <em>WAIC</em> (Widely Applicable Information Criterion), a closed-form estimate of elpd that uses every posterior draw. The second is <em>LOO-CV</em> (leave-one-out cross-validation), which estimates elpd by approximating what would happen if each observation were held out. LOO is now the recommended default; WAIC is an older approximation that often agrees but sometimes does not.</p>
<p>Both compute the same quantity in expectation: how well the fitted model would predict a new observation drawn from the same population. Higher elpd is better, like higher likelihood is better.</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">Three brms models, one LOO comparison</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>(loo)</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"># Three competing models for mpg</span></span>
<span class="cl">fit_a <span class="o"><-</span> <span class="nf">brm</span>(mpg <span class="o">~</span> wt, data <span class="o">=</span> mtcars, 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">1</span>, silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl">fit_b <span class="o"><-</span> <span class="nf">brm</span>(mpg <span class="o">~</span> wt <span class="o">+</span> hp, data <span class="o">=</span> mtcars, 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">1</span>, silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl">fit_c <span class="o"><-</span> <span class="nf">brm</span>(mpg <span class="o">~</span> wt <span class="o">*</span> hp, data <span class="o">=</span> mtcars, 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">1</span>, silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl"></span>
<span class="cl">loo_a <span class="o"><-</span> <span class="nf">loo</span>(fit_a)</span>
<span class="cl">loo_b <span class="o"><-</span> <span class="nf">loo</span>(fit_b)</span>
<span class="cl">loo_c <span class="o"><-</span> <span class="nf">loo</span>(fit_c)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">loo_compare</span>(loo_a, loo_b, loo_c)</span>
<span class="cl"><span class="c1">#> elpd_diff se_diff</span></span>
<span class="cl"><span class="c1">#> fit_c 0.0 0.0</span></span>
<span class="cl"><span class="c1">#> fit_b -1.6 2.1</span></span>
<span class="cl"><span class="c1">#> fit_a -3.0 3.4</span></span></div>
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<p>Walk through what just happened. We fit three nested-but-different models: <a class="auto-link" href="Simple-Linear-Regression-in-R.html" title="Linear Regression in R: Fit Your First Model With lm() and Understand Every Number">simple linear regression</a> on <code>wt</code>, plus <code>hp</code>, and with an interaction <code>wt * hp</code>. Each <code>loo()</code> call computed an elpd estimate using the full posterior of one fit, with no refitting and no held-out data needed from the user.</p>
<p><code>loo_compare()</code> ranked the three models by elpd. The best model has <code>elpd_diff = 0</code> (it is the reference). The other rows show how much worse each model is in elpd units, with a standard error on the difference (<code>se_diff</code>).</p>
<p>Now read the result. The interaction model <code>fit_c</code> is best, but only by 1.6 elpd units over <code>fit_b</code> with a standard error of 2.1. The standard rule of thumb is that an <code>elpd_diff</code> more than 4 standard errors below zero indicates a definitive preference; here <code>1.6 / 2.1 = 0.76</code>, well under 4.</p>
<p>The conclusion is that the data does not have enough evidence to prefer the interaction over the simpler <code>wt + hp</code> model. Pick the simpler one for parsimony.</p>
<p><img src="screenshots/Compare-Bayesian-Models-in-R-loo-flow.webp" alt="How LOO compares Bayesian models" class="img-responsive img-zoomable" loading="lazy" width="2248" height="664" /></p>
<p><em>Figure 1: The LOO comparison loop. Fit each model, compute <a class="auto-link" href="brms-in-R.html" title="brms in R: Bayesian Regression Without Writing a Single Line of Stan">loo()</a> on each, hand them to loo_compare(), and judge whether the elpd difference exceeds a few standard errors.</em></p>
<div class="callout callout-insight"><div class="callout-label">Key Insight</div><div class="callout-body"><strong>LOO comparison is one number with one standard error, not a <a class="auto-link" href="Statistical-Tests-in-R.html" title="Statistical Tests in R">p-value</a>.</strong> The output tells you how much better one model is <em>and</em> how confident you can be in that "better." Both numbers matter, and reporting the elpd difference alone is incomplete.</div></div>
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<p><strong>Try it:</strong> Fit a fourth model <code>mpg ~ wt + hp + cyl</code> and add it to the comparison. Where does it rank?</p>
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<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># fit_d <- brm(mpg ~ wt + hp + cyl, data = mtcars, ...)</span></span>
<span class="cl"><span class="c1"># loo_d <- loo(fit_d)</span></span>
<span class="cl"><span class="c1"># loo_compare(loo_a, loo_b, loo_c, loo_d)</span></span>
<span class="cl"><span class="c1">#> Expected: fit_d ranks similarly to fit_b or fit_c, the difference within sampling error</span></span></div>
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<details>
<summary>Click to reveal solution</summary>
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<div class="webr-editor" data-language="r"><span class="cl">fit_d <span class="o"><-</span> <span class="nf">brm</span>(mpg <span class="o">~</span> wt <span class="o">+</span> hp <span class="o">+</span> cyl, 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">1</span>, silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl">loo_d <span class="o"><-</span> <span class="nf">loo</span>(fit_d)</span>
<span class="cl"><span class="nf">loo_compare</span>(loo_a, loo_b, loo_c, loo_d)</span>
<span class="cl"><span class="c1">#> elpd_diff se_diff</span></span>
<span class="cl"><span class="c1">#> fit_c 0.0 0.0</span></span>
<span class="cl"><span class="c1">#> fit_d -0.7 1.6</span></span>
<span class="cl"><span class="c1">#> fit_b -1.6 2.1</span></span>
<span class="cl"><span class="c1">#> fit_a -3.0 3.4</span></span></div>
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<p>Adding <code>cyl</code> produces <code>fit_d</code>, which ranks just below <code>fit_c</code> by 0.7 elpd units (well within sampling error). The data cannot distinguish between the interaction model and the three-predictor model; either is a reasonable choice on predictive grounds. With only 32 cars, the safe call is the simpler model.</p>
</details>
</section>
<h2>What does LOO-CV actually compute?</h2>
<p>LOO-CV imagines holding out one observation, refitting the model on the remaining n-1 observations, and computing how well the refit predicts the held-out one. Repeat for each observation. Sum the per-observation log predictive densities; that sum is the elpd.</p>
<p>The naive implementation refits the model n times (once per held-out observation), which is enormously expensive. The <code>loo</code> package's trick is <em>importance sampling</em>: it estimates the leave-one-out elpd from the full-data posterior alone, without any refitting. The trick works when the posterior with one observation removed is similar enough to the full-data posterior that re-weighting samples is reliable.</p>
<p>The Pareto k diagnostic measures whether the trick is working. Each observation gets a <code>k</code> value. If <code>k < 0.5</code>, the importance sampling estimate for that observation is reliable.</p>
<p>If <code>k</code> is between 0.5 and 0.7, the estimate is acceptable. If <code>k > 0.7</code>, the estimate is unreliable and you should refit without that observation.</p>
<div class="webr-container" data-block-title="What is in a loo() object?">
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<div class="webr-editor" data-language="r"><span class="cl">loo_b</span>
<span class="cl"><span class="c1">#> Computed from 4000 by 32 log-likelihood matrix.</span></span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> Estimate SE</span></span>
<span class="cl"><span class="c1">#> elpd_loo -78.7 4.2</span></span>
<span class="cl"><span class="c1">#> p_loo 4.2 1.1</span></span>
<span class="cl"><span class="c1">#> looic 157.5 8.4</span></span>
<span class="cl"><span class="c1">#> ------</span></span>
<span class="cl"><span class="c1">#> Monte Carlo SE of elpd_loo is 0.0.</span></span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> All Pareto k estimates are good (k < 0.7).</span></span>
<span class="cl"><span class="c1">#> See help('pareto-k-diagnostic') for details.</span></span></div>
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<p>Walk through the report. <code>elpd_loo</code> is the expected log predictive density (higher is better, can be negative). <code>p_loo</code> is the <em>effective number of parameters</em>, an estimate of model complexity from the posterior. <code>looic</code> is <code>-2 * elpd_loo</code>, the LOO information criterion, which puts the metric on the AIC scale (lower is better) for people who prefer that.</p>
<p>The <code>MC SE of elpd_loo</code> is the Monte Carlo standard error from the finite number of posterior draws (here essentially zero, meaning we have plenty of draws). The "All Pareto k estimates are good" line confirms the importance sampling is reliable for every observation.</p>
<p>That last line is the safety net. When it says "k > 0.7" for one or more observations, you should not trust the elpd estimate. We come back to this in section six.</p>
<div class="callout callout-note"><div class="callout-label">Note</div><div class="callout-body"><strong><code>elpd_loo</code> itself is not interpretable in absolute terms.</strong> Whether <code>-78.7</code> is "good" depends on the data and the model. The number is meaningful only in <em>relative</em> comparison between models on the same data. That is why <code>loo_compare()</code> always reports differences, not absolute values.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Compute <code>loo()</code> on a single brms fit and inspect <code>loo$pointwise</code>, the per-observation elpd contributions. Which observation contributes the most negative elpd?</p>
<div class="webr-container" data-block-title="Your turn: per-observation elpd">
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<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># loo_b_pw <- loo_b$pointwise</span></span>
<span class="cl"><span class="c1"># Sort by elpd_loo column ascending; the most negative observation is first.</span></span>
<span class="cl"><span class="c1">#> Expected: one specific row of mtcars stands out as the hardest to predict</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">Per-observation elpd 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">loo_b_pw <span class="o"><-</span> loo_b<span class="o">$</span>pointwise</span>
<span class="cl">worst_3 <span class="o"><-</span> <span class="nf">order</span>(loo_b_pw[, <span class="s">"elpd_loo"</span>])[<span class="m">1</span><span class="o">:</span><span class="m">3</span>]</span>
<span class="cl"><span class="nf">data.frame</span>(</span>
<span class="cl"> car <span class="o">=</span> <span class="nf">rownames</span>(mtcars)[worst_3],</span>
<span class="cl"> mpg <span class="o">=</span> mtcars<span class="o">$</span>mpg[worst_3],</span>
<span class="cl"> pred_mpg_avg <span class="o">=</span> <span class="nf">round</span>(<span class="nf">rowMeans</span>(<span class="nf">posterior_predict</span>(fit_b))[worst_3], <span class="m">1</span>),</span>
<span class="cl"> elpd_loo <span class="o">=</span> <span class="nf">round</span>(loo_b_pw[worst_3, <span class="s">"elpd_loo"</span>], <span class="m">2</span>)</span>
<span class="cl">)</span>
<span class="cl"><span class="c1">#> car mpg pred_mpg_avg elpd_loo</span></span>
<span class="cl"><span class="c1">#> 1 Toyota Corolla 33.9 28.4 -4.97</span></span>
<span class="cl"><span class="c1">#> 2 Maserati Bora 15.0 13.4 -3.62</span></span>
<span class="cl"><span class="c1">#> 3 Lotus Europa 30.4 24.6 -3.30</span></span></div>
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<p>The Toyota Corolla contributes the most negative elpd (-4.97). Its observed mpg is 33.9, the highest in the dataset, and the model predicts 28.4. The Corolla is a known <a class="auto-link" href="Outlier-Treatment-With-R.html" title="Outlier Treatment with R">outlier</a> in mtcars (light, low-power, exceptionally efficient), and the model has trouble. Per-observation elpd is the LOO equivalent of looking for influential points in a frequentist regression.</p>
</details>
</section>
<h2>How do I run it in brms?</h2>
<p>The mechanics. brms exposes <code>loo()</code> directly: <code>loo(fit)</code> returns a loo object. Comparing models is <code>loo_compare(loo_a, loo_b)</code> or <code>loo_compare(loo_a, loo_b, loo_c)</code>. The objects work for any brms family; binomial, Poisson, Student-t, hierarchical, all the same syntax.</p>
<p>For very small datasets the LOO importance sampling is reliable. For very large datasets it is faster than refitting; for moderate ones it is fast enough to be the default.</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">Add LOO at the end of every brms fit</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">fit_b <span class="o"><-</span> <span class="nf">brm</span>(mpg <span class="o">~</span> wt <span class="o">+</span> hp, mtcars, 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">1</span>, silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl">loo_b <span class="o"><-</span> <span class="nf">loo</span>(fit_b)</span>
<span class="cl">loo_b</span>
<span class="cl"><span class="c1">#> Estimate SE</span></span>
<span class="cl"><span class="c1">#> elpd_loo -78.7 4.2</span></span>
<span class="cl"><span class="c1">#> p_loo 4.2 1.1</span></span>
<span class="cl"><span class="c1">#> looic 157.5 8.4</span></span>
<span class="cl"><span class="c1">#> All Pareto k estimates are good (k < 0.7).</span></span></div>
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<p>Walk through what we just did. <code>loo()</code> ran on the fitted brmsfit object. It used the full 4000-draw posterior to compute per-observation elpd and aggregated them.</p>
<p>The output reports <code>elpd_loo</code>, the effective number of parameters <code>p_loo</code>, and <code>looic</code>. The Pareto k summary at the bottom is the one piece you must not skip.</p>
<p>For comparing models with different families on the same y (Normal vs Student-t, for instance), the comparison still works because elpd is on the log-density scale of y; the family difference is folded into the per-observation log-likelihood that <code>loo()</code> reads.</p>
<div class="webr-container" data-block-title="Comparing different families on the same outcome">
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<div class="webr-editor" data-language="r"><span class="cl">fit_normal <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">1</span>, silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl">fit_student <span class="o"><-</span> <span class="nf">brm</span>(mpg <span class="o">~</span> wt <span class="o">+</span> hp, mtcars, family <span class="o">=</span> <span class="nf">student</span>(),</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">1</span>, silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">loo_compare</span>(<span class="nf">loo</span>(fit_normal), <span class="nf">loo</span>(fit_student))</span>
<span class="cl"><span class="c1">#> elpd_diff se_diff</span></span>
<span class="cl"><span class="c1">#> fit_normal 0.0 0.0</span></span>
<span class="cl"><span class="c1">#> fit_student -0.4 1.5</span></span></div>
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<p>Walk through what we did. Both fits use the same formula <code>mpg ~ wt + hp</code> but different families. The Normal family fits slightly better in elpd terms (by 0.4 with se 1.5), but the difference is well within sampling error.</p>
<p>For mtcars the Normal family is fine. For datasets with heavy tails or visible outliers, the same comparison would show Student-t winning by 4 or more elpd units. This is how you make the data-driven choice between likelihood families without committing to one a priori.</p>
<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong>Cache loo objects with the model.</strong> Computing loo from a fresh brms fit takes a few seconds; computing it for several models in a comparison is repeated work. The brms convention is to add <code>fit$loo <- loo(fit)</code> (or save the object alongside) so you do not recompute.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Compute LOO for both <code>fit_normal</code> and <code>fit_student</code>. Print only the elpd estimates and standard errors as a tidy two-row data frame.</p>
<div class="webr-container" data-block-title="Your turn: tidy loo output">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: tidy loo output</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Build a data.frame with model name, elpd_loo estimate, elpd_loo SE</span></span>
<span class="cl"><span class="c1"># Tip: loo_x$estimates['elpd_loo', 'Estimate']</span></span></div>
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<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">data.frame</span>(</span>
<span class="cl"> model <span class="o">=</span> <span class="nf">c</span>(<span class="s">"normal"</span>, <span class="s">"student"</span>),</span>
<span class="cl"> elpd_loo <span class="o">=</span> <span class="nf">c</span>(<span class="nf">loo</span>(fit_normal)<span class="o">$</span>estimates[<span class="s">"elpd_loo"</span>, <span class="s">"Estimate"</span>],</span>
<span class="cl"> <span class="nf">loo</span>(fit_student)<span class="o">$</span>estimates[<span class="s">"elpd_loo"</span>, <span class="s">"Estimate"</span>]),</span>
<span class="cl"> elpd_loo_SE <span class="o">=</span> <span class="nf">c</span>(<span class="nf">loo</span>(fit_normal)<span class="o">$</span>estimates[<span class="s">"elpd_loo"</span>, <span class="s">"SE"</span>],</span>
<span class="cl"> <span class="nf">loo</span>(fit_student)<span class="o">$</span>estimates[<span class="s">"elpd_loo"</span>, <span class="s">"SE"</span>])</span>
<span class="cl">) <span class="o">|></span> <span class="nf">round</span>(<span class="m">2</span>)</span>
<span class="cl"><span class="c1">#> model elpd_loo elpd_loo_SE</span></span>
<span class="cl"><span class="c1">#> 1 normal -78.74 4.21</span></span>
<span class="cl"><span class="c1">#> 2 student -79.13 4.45</span></span></div>
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<p>Both models have similar elpd (-78.7 vs -79.1) with similar SEs. The student-t model is slightly worse in elpd but well within the SE of the comparison. For mtcars, neither family is wrong.</p>
</details>
</section>
<h2>How do I read elpd_diff and the standard error?</h2>
<p><code>loo_compare()</code> returns a table with two columns. <code>elpd_diff</code> is the difference in elpd between each model and the best (the best gets <code>elpd_diff = 0</code>). <code>se_diff</code> is the standard error of that difference.</p>
<p>The rule of thumb: a model is <em>definitively</em> worse if <code>|elpd_diff| > 4 * se_diff</code>. Below that threshold, the comparison is uncertain. Many published Bayesian analyses use looser thresholds (2 SE for a "weak preference," 4 SE for "strong"); both are defensible.</p>
<div class="webr-container" data-block-title="Reading loo_compare output">
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<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">loo_compare</span>(loo_a, loo_b, loo_c)</span>
<span class="cl"><span class="c1">#> elpd_diff se_diff</span></span>
<span class="cl"><span class="c1">#> fit_c 0.0 0.0</span></span>
<span class="cl"><span class="c1">#> fit_b -1.6 2.1</span></span>
<span class="cl"><span class="c1">#> fit_a -3.0 3.4</span></span>
<span class="cl"></span>
<span class="cl"><span class="nf">abs</span>(<span class="nf">c</span>(b <span class="o">=</span> <span class="m">-1.6</span> <span class="o">/</span> <span class="m">2.1</span>, a <span class="o">=</span> <span class="m">-3.0</span> <span class="o">/</span> <span class="m">3.4</span>))</span>
<span class="cl"><span class="c1">#> b a</span></span>
<span class="cl"><span class="c1">#> 0.762 0.882</span></span></div>
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<p>Walk through the decision. <code>fit_b</code> is 1.6 elpd units worse than <code>fit_c</code> with se 2.1; ratio 0.76. <code>fit_a</code> is 3.0 worse with se 3.4; ratio 0.88.</p>
<p>Both ratios are well below 4, so the data does not decisively prefer <code>fit_c</code> over the alternatives. Reporting "fit_c is best" without the ratio context would mislead. The honest framing is that fit_c has the highest elpd, but the gap to fit_b and fit_a is within 1 SE; the data cannot reliably distinguish these models.</p>
<p>For policy or scientific decisions, you typically want a stronger signal. Here is what 4-SE evidence looks like:</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">A clearer LOO comparison</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">fit_null <span class="o"><-</span> <span class="nf">brm</span>(mpg <span class="o">~</span> <span class="m">1</span>, mtcars, 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">1</span>, silent <span class="o">=</span> <span class="m">2</span>)</span>
<span class="cl">loo_null <span class="o"><-</span> <span class="nf">loo</span>(fit_null)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">loo_compare</span>(loo_b, loo_null)</span>
<span class="cl"><span class="c1">#> elpd_diff se_diff</span></span>
<span class="cl"><span class="c1">#> fit_b 0.0 0.0</span></span>
<span class="cl"><span class="c1">#> fit_null -22.4 6.1</span></span></div>
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<p>Walk through what changed. The null model (intercept only) has 22.4 fewer elpd units than <code>fit_b</code> with an SE of 6.1; the ratio is <code>22.4 / 6.1 = 3.7</code>, just under our 4-SE threshold. The signal is strong: <code>fit_b</code> is much better at predicting new data than the null model.</p>
<p>This is the kind of comparison where you can confidently report that the regression model substantially outperforms the no-predictor baseline. Reporting the ratio alongside the elpd_diff itself makes the decision auditable.</p>
<p><img src="screenshots/Compare-Bayesian-Models-in-R-criteria.webp" alt="Bayesian model comparison criteria" class="img-responsive img-zoomable" loading="lazy" width="1034" height="986" /></p>
<p><em>Figure 2: Three model-comparison criteria. AIC uses an MLE point and is wrong for Bayesian fits. WAIC uses the full posterior in closed form. LOO-CV is the modern recommended default with honest SE and a built-in reliability diagnostic.</em></p>
<div class="callout callout-note"><div class="callout-label">Note</div><div class="callout-body"><strong><code>elpd_diff > 0</code> is impossible by construction.</strong> The best model always has 0; everything else is below. Newcomers sometimes look for a positive elpd_diff and conclude something is wrong; that is just the convention.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Compute <code>loo_compare()</code> on <code>fit_b</code> and <code>fit_null</code> (the intercept-only model from above). Report the ratio <code>|elpd_diff| / se_diff</code>.</p>
<div class="webr-container" data-block-title="Your turn: compute the ratio">
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<div class="webr-editor" data-language="r"><span class="cl"><span class="c1"># Use loo_compare(loo_b, loo_null) and pull elpd_diff and se_diff for fit_null</span></span>
<span class="cl"><span class="c1"># Compute abs(elpd_diff) / se_diff</span></span></div>
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<div class="webr-editor" data-language="r"><span class="cl">cmp <span class="o"><-</span> <span class="nf">loo_compare</span>(loo_b, loo_null)</span>
<span class="cl">cmp</span>
<span class="cl"><span class="c1">#> elpd_diff se_diff</span></span>
<span class="cl"><span class="c1">#> fit_b 0.0 0.0</span></span>
<span class="cl"><span class="c1">#> fit_null -22.4 6.1</span></span>
<span class="cl"><span class="nf">abs</span>(cmp[<span class="s">"fit_null"</span>, <span class="s">"elpd_diff"</span>]) <span class="o">/</span> cmp[<span class="s">"fit_null"</span>, <span class="s">"se_diff"</span>]</span>
<span class="cl"><span class="c1">#> [1] 3.672</span></span></div>
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<p>The ratio is 3.67, strong evidence that <code>fit_b</code> predicts better. By the 4-SE rule the comparison is borderline definitive; by the more common 2-SE threshold it is solidly definitive. Either way, the regression model is much better than the null.</p>
</details>
</section>
<h2>What does WAIC do, and when is it different from LOO?</h2>
<p>WAIC (Widely Applicable Information Criterion) is a closed-form estimate of elpd that uses every posterior draw plus a complexity penalty. It does not need importance sampling and has no Pareto k diagnostic. It is faster than LOO when the dataset is large or the importance-sampling step in LOO struggles.</p>
<p>For most well-fitting models, WAIC and LOO agree. The two differ when:</p>
<ol>
<li>The model has heavy-tailed posteriors that cause Pareto k warnings in LOO.</li>
<li>Some observations are highly influential (outliers, or important leverage points).</li>
<li>The dataset is very large and importance sampling becomes numerically tricky.</li>
</ol>
<p>LOO is generally preferred because it has the diagnostic; WAIC silently fails when the elpd approximation is bad. Running both as a cross-check is a 30-second workflow addition.</p>
<div class="webr-container" data-block-title="WAIC vs LOO on the same fit">
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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">WAIC vs LOO on the same fit</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">waic_b <span class="o"><-</span> <span class="nf">waic</span>(fit_b)</span>
<span class="cl">waic_b</span>
<span class="cl"><span class="c1">#> Computed from 4000 by 32 log-likelihood matrix.</span></span>
<span class="cl"><span class="c1">#> Estimate SE</span></span>
<span class="cl"><span class="c1">#> elpd_waic -78.7 4.2</span></span>
<span class="cl"><span class="c1">#> p_waic 4.2 1.1</span></span>
<span class="cl"><span class="c1">#> waic 157.5 8.5</span></span>
<span class="cl"></span>
<span class="cl"><span class="nf">loo_compare</span>(loo_b, waic_b)</span>
<span class="cl"><span class="c1">#> elpd_diff se_diff</span></span>
<span class="cl"><span class="c1">#> waic_b 0.0 0.0</span></span>
<span class="cl"><span class="c1">#> loo_b -0.0 0.0</span></span></div>
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<p>Walk through what just happened. <code>waic_b</code> returned essentially the same <code>elpd_waic</code> (-78.7) as <code>loo_b</code> returned for <code>elpd_loo</code> (-78.7), and the same SE (4.2 vs 4.2). The LOO and WAIC comparisons differ by 0.0; for this fit they agree exactly.</p>
<p>When they disagree, it is often a sign of <a class="auto-link" href="Posterior-Predictive-Checks-in-R.html" title="Posterior Predictive Checks in R: The 5-Minute Way to Catch a Broken Model">model misspecification</a>. WAIC understates uncertainty in the presence of <a class="auto-link" href="Regression-Diagnostics-in-R.html" title="Regression Diagnostics in R: 5 Plots That Reveal Model Violations Instantly">influential observations</a>; LOO over-corrects. If the two differ by more than 1 elpd unit, look at the LOO Pareto k values and rerun with <code>reloo</code> (next section) for the bad observations.</p>
<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong>Use <code>loo()</code> by default, run <code>waic()</code> as a cross-check on important comparisons.</strong> When the two agree (within 1 elpd unit), you can trust either. When they disagree, the model has unusual structure and the LOO Pareto k diagnostic will tell you which observations are the source.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Compute <code>waic()</code> for the three mtcars models from earlier and confirm the ranking matches the LOO ranking.</p>
<div class="webr-container" data-block-title="Your turn: WAIC ranking">
<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">Your turn: WAIC ranking</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"># Compute waic_a, waic_b, waic_c and pass to loo_compare()</span></span>
<span class="cl"><span class="c1"># Compare to the loo_compare(loo_a, loo_b, loo_c) ranking</span></span></div>
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<div class="webr-editor" data-language="r"><span class="cl">waic_a <span class="o"><-</span> <span class="nf">waic</span>(fit_a)</span>
<span class="cl">waic_b <span class="o"><-</span> <span class="nf">waic</span>(fit_b)</span>
<span class="cl">waic_c <span class="o"><-</span> <span class="nf">waic</span>(fit_c)</span>
<span class="cl"><span class="nf">loo_compare</span>(waic_a, waic_b, waic_c)</span>
<span class="cl"><span class="c1">#> elpd_diff se_diff</span></span>
<span class="cl"><span class="c1">#> fit_c 0.0 0.0</span></span>
<span class="cl"><span class="c1">#> fit_b -1.6 2.1</span></span>
<span class="cl"><span class="c1">#> fit_a -3.0 3.4</span></span></div>
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<p>WAIC ranking matches the LOO ranking exactly (<code>fit_c</code> > <code>fit_b</code> > <code>fit_a</code>), with the same elpd differences and standard errors. For this dataset the two methods agree, which is the common case. When they disagree, look at LOO's Pareto k diagnostic.</p>
</details>
</section>
<h2>When does LOO break (Pareto-k warnings) and what do you do?</h2>
<p>LOO's importance sampling assumes that for each held-out observation, the full-data posterior is similar to the leave-one-out posterior. When that assumption breaks (an influential observation, an outlier, a model that is highly sensitive to one point), LOO returns a Pareto k value above 0.7 for that observation and prints a warning.</p>
<p>The fix has two paths.</p>