Beta model
Description
Beta regression is used for outcomes on the [0, 1] interval.
Definition
For continuous outcome \(y\) bounded [0, 1] and predictors \(x\), the model is:
\[ \begin{align} y_i \sim \text{beta}(\text{logit}^{-1}(\mu_i), \kappa) \\ \mu_i = \alpha + \beta \cdot x_i \\ \end{align} \]
The logit function is the quantile function associated with the standard logistic distribution. Its inverse maps \(\mu_i\) from the real line into a value within the interval (0, 1).
Parameters needing priors
- \(\alpha\) (intercept)
- \(\beta\) (predictor weights)
- \(\kappa\) (concentration parameter)
The larger the value of \(\kappa\), the more concentrated the distribution is around the mean. This parameter is also named the precision parameter or the sample size parameter. Greek letters \(\nu\) and \(\phi\) are also commonly used to refer to this parameter.
Prior for \(\kappa\)
For weakly informative prior we can use a gamma distribution with low probability mass on very small values of \(\kappa\), such priors disfavor u-shaped beta distributions which are generally less common in real-world data. For instance, we could choose \(\text{gamma}(4, 0.1)\) as discussed by Solomon Kurz, which place most of the prior mass on the double-digit range with long right tail allowing for greater concentrations if needed.