## 11.2 Bernoulli Distribution, Logit Parameterization

Stan also supplies a direct parameterization in terms of a logit-transformed chance-of-success parameter. This parameterization is more numerically stable if the chance-of-success parameter is on the logit scale, as with the linear predictor in a logistic regression.

### 11.2.1 Probability Mass Function

If $$\alpha \in \mathbb{R}$$, then for $$y \in \{0,1\}$$, $\text{BernoulliLogit}(y~|~\alpha) = \text{Bernoulli}(y | \text{logit}^{-1}(\alpha)) = \left\{ \begin{array}{ll} \text{logit}^{-1}(\alpha) & \text{if } y = 1, \text{ and} \\ 1 - \text{logit}^{-1}(\alpha) & \text{if } y = 0. \end{array} \right.$

### 11.2.2 Sampling Statement

y ~ bernoulli_logit(alpha)

Increment target log probability density with bernoulli_logit_lpmf( y | alpha) dropping constant additive terms.

### 11.2.3 Stan Functions

real bernoulli_logit_lpmf(ints y | reals alpha)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha)

R bernoulli_logit_rng(reals alpha)
Generate a Bernoulli variate with chance of success $$\text{logit}^{-1}(\alpha)$$; may only be used in generated quantities block. For a description of argument and return types, see section vectorized PRNG functions.