4.6 Summary

Chapter 4 in review

Chapter 4 introduced the linear model, the workhorse of statistics. We started with the Gaussian distribution and explained why it is a natural default (central limit theorem, maximum entropy). We then built a language for describing models that separates the likelihood, the linear model, and the priors.

We fit a Gaussian model of height (no predictor) and then extended it to a linear regression with weight as a predictor. We learned the importance of prior predictive simulation, centering predictors, and visualizing posterior uncertainty. Finally, we explored polynomial regression for nonlinear relationships and saw the dangers of overfitting.

Key models from Chapter 4

• Gaussian model: $h_i \sim \text{Normal}(\mu, \sigma)$
• Linear regression: $h_i \sim \text{Normal}(\alpha + \beta x_i, \sigma)$
• Polynomial regression: $h_i \sim \text{Normal}(\alpha + \beta_1 x_i + \beta_2 x_i^2, \sigma)$

Key lessons

1. Choose priors carefully: Use prior predictive simulation to check what the priors imply.
2. Center predictors: Makes the intercept interpretable and improves computation.
3. Interpret coefficients in context: The slope is the expected change per unit of the predictor.
4. Check the model: Use posterior predictive simulations to verify the model fits.
5. Beware overfitting: Use the simplest model that captures the important features.