## 13.6 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.

### 13.6.1 Probability Mass Function

If $$\alpha \in \mathbb{R}$$, then for $$n \in \mathbb{N}$$, $\text{PoissonLog}(n|\alpha) = \frac{1}{n!} \, \exp \left(n\alpha - \exp(\alpha) \right).$

### 13.6.2 Sampling Statement

n ~ poisson_log(alpha)

Increment target log probability density with poisson_log_lpmf(n | alpha) dropping constant additive terms.

### 13.6.3 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.