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In the definition of the exponential distribution with the scale parameter beta, the probability density function should be
\begin{equation*}
\text{Exponential}(y|\beta) = \frac{1}{\beta} , \exp ( -\frac{y}{\beta} ) . \end{equation*}
My bad, I thought that the parameterization was with the scale, and not the inverse scale, and that there was a mistake in the pdf definition. Yes, the parameter should be \lambda everywhere. Maybe it's worth writing that it is also the rate parameter to avoid confusion.
In the definition of the exponential distribution with the scale parameter beta, the probability density function should be
\begin{equation*}
\text{Exponential}(y|\beta) = \frac{1}{\beta} , \exp ( -\frac{y}{\beta} ) . \end{equation*}
https://mc-stan.org/docs/functions-reference/positive_continuous_distributions.html#exponential-distribution
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