Multiparameter Models

Summary

Chapter 3 of BDA3 extends Bayesian inference to models with multiple parameters. The key technique is averaging over “nuisance parameters” to obtain marginal posteriors for quantities of interest.

Marginalizing Over Nuisance Parameters

For parameters $({\theta }_1,{\theta }_2)$ where ${\theta }_2$ is a nuisance parameter:

$$p({\theta }_1\mid y)=\int p({\theta }_1,{\theta }_2\mid y)d{\theta }_2$$

This is conceptually clean but often analytically intractable — motivating computational methods in Part III.

Normal Data with Unknown Mean and Variance

With $y_i\sim N(\mu ,{\sigma }^2)$ and a noninformative prior $p(\mu ,{\sigma }^2)\propto {\sigma }^{-2}$:

1. Marginal posterior for ${\sigma }^2$: scaled inverse-${\chi }^2$ distribution
2. Conditional posterior: $\mu \mid {\sigma }^2,y\sim N(\bar{y},{\sigma }^2/n)$
3. Marginal posterior for $\mu$: $\frac{\mu -\bar{y}}{s/\sqrt{n}}\mid y\sim t_{n-1}$ — recovering the familiar $t$-distribution

Other Models Covered

• Multinomial model: Dirichlet-multinomial conjugacy for categorical data
• Multivariate normal: conjugate analysis with known/unknown covariance
• Bioassay example: logistic regression with two parameters — non-conjugate, requires grid or simulation methods

Key Insight

Tip

With simulation, multiparameter problems reduce to repeated single-parameter problems: draw from the joint posterior, then examine marginals by simply looking at individual components.

See Also

• Single-Parameter Models — the building blocks
• Asymptotics and Frequentist Connections — large-sample behavior
• Bayesian Linear Regression — the full regression setting