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aloctavodia authored Sep 12, 2024
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Expand Up @@ -27,12 +27,12 @@ The [logit](https://en.wikipedia.org/wiki/Logit) function is the quantile functi
- $\beta$ (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 parameters. Greek letters $\nu$ and $\phi$ are also commonly used to refer to this 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 disfavors 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](https://solomonkurz.netlify.app/blog/2023-06-25-causal-inference-with-beta-regression/), wich place most of the prior mass on the double-digit range with long right tail allowing for greater concentrations if needed.
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](https://solomonkurz.netlify.app/blog/2023-06-25-causal-inference-with-beta-regression/), which place most of the prior mass on the double-digit range with long right tail allowing for greater concentrations if needed.


## See also
Expand All @@ -42,4 +42,4 @@ For weakly informative prior we can use a gamma distribution with low probabilit
- [Stan Functions Reference](https://mc-stan.org/docs/functions-reference/continuous_distributions_on_0_1.html#beta-distribution)
- [rstanarm manual](https://mc-stan.org/rstanarm/articles/betareg.html)
- [bambi manual](https://bambinos.github.io/bambi/notebooks/beta_regression.html)
- [Andrew Wheiss' blog](https://www.andrewheiss.com/blog/2021/11/08/beta-regression-guide/)
- [Andrew Wheiss' blog](https://www.andrewheiss.com/blog/2021/11/08/beta-regression-guide/)

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