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13.6 Poisson Distribution, Log Parameterization

Stan also provides a parameterization of the Poisson using the log rate α=logλ 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 αR, then for nN, PoissonLog(n|α)=1n!exp(nαexp(α)).

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 30log2. For a description of argument and return types, see section vectorized function signatures.