Missing Data Models

Summary

Chapter 18 of BDA3 presents the Bayesian framework for handling missing data. Multiple imputation — drawing multiple plausible completions of the data from the posterior predictive distribution — propagates missing-data uncertainty into final inferences.

Missing Data Mechanisms

• MCAR (Missing Completely At Random): missingness independent of all data
• MAR (Missing At Random): missingness depends only on observed values — mechanism is ignorable
• MNAR (Missing Not At Random): missingness depends on the missing values — requires explicit modeling of the mechanism

Multiple Imputation

1. Draw $M$ completed datasets from $p(y_{\text{mis}}\mid y_{\text{obs}})$
2. Analyze each completed dataset separately
3. Combine results using Rubin’s rules:
   • Point estimate: $\bar{Q}=\frac{1}{M}{\sum }_{m=1}^M{\hat{Q}}_m$
   • Variance: $T=\bar{U}+(1+1/M)B$ where $\bar{U}$ is within-imputation variance and $B$ is between-imputation variance

Tip

In a fully Bayesian analysis, missing data are simply additional unknown parameters — they are sampled alongside model parameters in each MCMC iteration. Multiple imputation approximates this for non-Bayesian analyses.

Key Applications

• Polls with missing demographic data: imputing covariates for poststratification
• Counted data: handling partially observed counts (e.g., election data with missing precincts)

See Also

• Data Collection Models — ignorability conditions (Ch 8)
• Hierarchical Models — hierarchical imputation models
• Missing Data - Statistical Rethinking — DAG-based treatment of missing data mechanisms from Statistical Rethinking (companion note)
• MCMC Basics — in a fully Bayesian analysis, missing values are sampled alongside model parameters in each MCMC iteration