14.3 Summary
Chapter 14 in review
Chapter 14 showed that measurement error and missing data are not problems to ignore but opportunities to model. By treating unknown values as parameters, Bayesian models propagate uncertainty naturally.
We covered:
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Measurement error (Section 14.1): Noisy predictors attenuate the slope. The Bayesian model treats true values as latent variables, estimating them alongside the regression parameters. This corrects the attenuation and propagates uncertainty.
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Missing data (Section 14.2): Missing values can bias results. MCAR is harmless but rare; MAR requires imputation; MNAR requires a model of missingness. Bayesian imputation treats missing values as parameters, propagating uncertainty into the estimates of interest.
Key formulas
- Measurement error:
- Attenuation:
- Imputation:
Key lessons
- Measurement error attenuates relationships — ignoring it is a modeling decision.
- Missing data are a modeling opportunity — treat unknowns as parameters.
- MCAR is safe but rare; MAR needs imputation; MNAR needs a missingness model.
- Bayesian imputation propagates uncertainty naturally into the posterior.
- Ignoring uncertainty in data is itself a modeling decision, and usually a bad one.
Unknowns are parameters. The deep lesson of Chapter 14 is that anything unknown — whether a measurement error, a missing value, or a latent state — can be treated as a parameter and estimated. This is the Bayesian way: instead of ignoring uncertainty, model it. Instead of dropping data, impute it. Instead of pretending measurements are exact, model the measurement process. Each of these choices adds parameters, but it also adds honesty. The posterior will be wider, but it will be more accurate. A narrow but biased estimate is worse than a wide but honest one. This is the essence of Bayesian inference: the goal is not to be certain, but to be honest about uncertainty.