4. Linear Models

Overview

• 4.1 Why normal distributions are normal

  • 4.1.1 Normal by addition

  • 4.1.2 Normal by multiplication

  • 4.1.3 Normal by log-multiplication

  • 4.1.4 Using Gaussian distributions

    • 4.1.4.1 Ontological justification

    • 4.1.4.2 Epistemological justification

      • Gaussian distribution

• 4.2 A language for describing models

  • 4.2.1 Re-describing the globe tossing model

    • From model definition to Bayes’ theorem

• 4.3 A Gaussian model of height

  • 4.3.1 The data

    • Data frames

    • Index magic

  • 4.3.2 The model

    • Independent and identically distributed

    • A farewell to epsilon

    • Model definition to Bayes’ theorem again

  • 4.3.3 Grid approximation of the posterior distribution

  • 4.3.4 Sampling from the posterior

    • Sample size and the normality of σ’s posterior

  • 4.3.5 Fitting the model with map

    • Start values for map

    • How strong is a prior?

  • 4.3.6 Sampling from a map fit

    • Under the hood with multivariate sampling

    • Getting σ right

• 4.4 Adding a predictor

  • What is “regression”?

  • 4.4.1 The linear model strategy

    • 4.4.1.1 Likelihood

    • 4.4.1.2 Linear model

      • Nothing special or natural about linear models

      • Units and regression models

    • 4.4.1.3 Priors

      • What’s the correct prior?

  • 4.4.2 Fitting the model

    • Everything that depends upon parameters has a posterior distribution

    • Embedding linear models

  • 4.4.3 Interpreting the model fit

    • What do parameters mean?

    • 4.4.3.1 Tables of estimates

    • 4.4.3.2 Plotting posterior inference against the data

    • 4.4.3.3 Adding uncertainty around the mean

    • 4.4.3.4 Plotting regression intervals and contours

      • Overconfident confidence intervals

      • How link works

    • 4.4.3.5 Prediction intervals

      • Two kinds of uncertainty

      • Rolling your own sim

• 4.5 Polynomial regression

  • Linear, additive, funky

  • Converting back to natural scale

• 4.6 Summary