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## 14.3 Negative Binomial Distribution (log alternative parameterization)

Related to the parameterization in section negative binomial, alternative parameterization, the following parameterization uses a log mean parameter $$\eta = \log(\mu)$$, defined for $$\eta \in \mathbb{R}$$, $$\phi \in \mathbb{R}^+$$, so that for $$n \in \mathbb{N}$$, $\text{NegBinomial2Log}(n \, | \, \eta, \phi) = \text{NegBinomial2}(n | \exp(\eta), \phi).$ This alternative may be used for sampling, as a function, and for random number generation, but as of yet, there are no CDFs implemented for it. This is especially useful for log-linear negative binomial regressions.

### 14.3.1 Sampling Statement

n ~ neg_binomial_2_log(eta, phi)

Increment target log probability density with neg_binomial_2_log_lupmf(n | eta, phi).

### 14.3.2 Stan Functions

real neg_binomial_2_log_lpmf(ints n | reals eta, reals phi)
The log negative binomial probability mass of n given log-location eta and inverse overdispersion parameter phi.

real neg_binomial_2_log_lupmf(ints n | reals eta, reals phi)
The log negative binomial probability mass of n given log-location eta and inverse overdispersion parameter phi dropping constant additive terms.

R neg_binomial_2_log_rng(reals eta, reals phi)
Generate a negative binomial variate with log-location eta and inverse overdispersion control phi; may only be used in transformed data and generated quantities blocks. eta must be less than $$29 \log 2$$. For a description of argument and return types, see section vectorized function signatures.