Normal model
Description
Linear regression with normally distributed residuals.
Definition
For continuous unbounded outcome \(y\) and predictors \(x\), the model is:
\[ \begin{align} y_i \sim \text{normal}(\eta_i, \sigma) \\ \eta_i = \alpha + \beta \cdot x_i \end{align} \]
Parameters needing priors
- \(\alpha\) (intercept)
- \(\beta\) (predictor weights)
- \(\sigma\) (residual standard deviation)
Prior for \(\alpha\)
Weakly informative data-adaptive prior
Gelman, Hill, and Vehtari (2020) Chapter 9
Centered intercept (expected value of \(y\) when predictors are set to mean values of observed data)
\[ \begin{align} \alpha_{\text{centered}} \sim \text{normal}(\text{mean}(y), 2.5 \text{SD}(y)) \\ \end{align} \]
Prior for \(\beta\)
Weakly informative data-adaptive normal prior
Gelman, Hill, and Vehtari (2020) Chapter 9 describes a data-adaptive normal prior
\[ \begin{align} \beta_k \sim \text{normal}(0, 2.5 \text{SD}(y)/\text{SD}(x_k)). \end{align} \]
Prior for \(\sigma\)
Weakly informative data-adaptive exponential prior
Gelman, Hill, and Vehtari (2020) Chapter 9 describes a data-adaptive exponential prior
\[ \sigma \sim \text{exponential}(1/\text{SD}(y)) \]