14.6 Poisson Distribution, Log Parameterization
Stan also provides a parameterization of the Poisson using the log rate \(\alpha = \log \lambda\) as a parameter. This is useful for log-linear Poisson regressions so that the predictor does not need to be exponentiated and passed into the standard Poisson probability function.
14.6.1 Probability Mass Function
If \(\alpha \in \mathbb{R}\), then for \(n \in \mathbb{N}\), \[ \text{PoissonLog}(n|\alpha) = \frac{1}{n!} \, \exp \left(n\alpha - \exp(\alpha) \right). \]
14.6.2 Sampling Statement
n ~ poisson_log(alpha)
Increment target log probability density with poisson_log_lpmf(n | alpha)
dropping constant additive terms.
14.6.3 Stan Functions
real poisson_log_lpmf(ints n | reals alpha)
The log Poisson probability mass of n given log rate alpha
R poisson_log_rng(reals alpha)
Generate a Poisson variate with log rate alpha; may only be used in
transformed data and generated quantities blocks. alpha must be less than
\(30 \log 2\). For a description of argument and return types, see section
vectorized function signatures.