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12.2 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.

12.2.1 Probability Mass Function

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

12.2.2 Sampling Statement

y ~ bernoulli_logit(alpha)

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

12.2.3 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 transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.