Add gamma regression #25
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| $$ | ||
| \begin{align} | ||
| y_i \sim \text{gamma}(s, \alpha/exp(\eta)) \\ | ||
| \eta_i = \alpha + \beta \cdot x_i | ||
| \end{align} | ||
| $$ |
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The Gamma distribution can be a tricky one given the different parameterizations.
In PyMC/Bambi/PreliZ we have (alpha, beta) and (mu, sigma) see here for details https://preliz.readthedocs.io/en/latest/examples/gallery/gamma.html
Usually, we use the mu, sigma parametrization like this
\begin{align}
y_i \sim \text{gamma}(\text{mu}=\mu_i, \text{sigma}=\frac{\mu_i}{\alpha^{0.5}}) \
\mu_i = \exp(\eta_i)\
\eta_i = \beta_0 + \beta_1 \cdot x_i
\end{align}
In brms, I think we have (mu, alpha) and Stan use (alpha, beta) [the same as the PyMC's default]
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Good point, there are clear typos here too. However the issue of different parameterizations is something we will run into in various places, also when recommending priors (e.g. exponential prior). I will open an issue about handling different parametrisations in general
WIP