# Binary Distributions

Binary probability distributions have support on $$\{0,1\}$$, where 1 represents the value true and 0 the value false.

## Bernoulli distribution

### Probability mass function

If $$\theta \in [0,1]$$, then for $$y \in \{0,1\}$$, $\begin{equation*} \text{Bernoulli}(y~|~\theta) = \left\{ \begin{array}{ll} \theta & \text{if } y = 1, \text{ and} \\ 1 - \theta & \text{if } y = 0. \end{array} \right. \end{equation*}$

### Distribution statement

y ~ bernoulli(theta)

Increment target log probability density with bernoulli_lupmf(y | theta).

Available since 2.0

### Stan Functions

real bernoulli_lpmf(ints y | reals theta)
The log Bernoulli probability mass of y given chance of success theta

Available since 2.12

real bernoulli_lupmf(ints y | reals theta)
The log Bernoulli probability mass of y given chance of success theta dropping constant additive terms

Available since 2.25

real bernoulli_cdf(ints y | reals theta)
The Bernoulli cumulative distribution function of y given chance of success theta

Available since 2.0

real bernoulli_lcdf(ints y | reals theta)
The log of the Bernoulli cumulative distribution function of y given chance of success theta

Available since 2.12

real bernoulli_lccdf(ints y | reals theta)
The log of the Bernoulli complementary cumulative distribution function of y given chance of success theta

Available since 2.12

ints bernoulli_rng(reals theta)
Generate a Bernoulli variate with chance of success theta or an array of Bernoulli variates given an array of thetas of the same dimensions; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.

Available since 2.18

## 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.

### Probability mass function

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

### Distribution statement

y ~ bernoulli_logit(alpha)

Increment target log probability density with bernoulli_logit_lupmf(y | alpha).

Available since 2.0

### Stan Functions

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

Available since 2.12

real bernoulli_logit_lupmf(ints y | reals alpha)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha) dropping constant additive terms

Available since 2.25

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.

Available since 2.18

## Bernoulli-logit generalized linear model (Logistic Regression)

Stan also supplies a single function for a generalized linear model with Bernoulli distribution and logit link function, i.e. a function for a logistic regression. This provides a more efficient implementation of logistic regression than a manually written regression in terms of a Bernoulli distribution and matrix multiplication.

### Probability mass function

If $$x\in \mathbb{R}^{n\cdot m}, \alpha \in \mathbb{R}^n, \beta\in \mathbb{R}^m$$, then for $$y \in {\{0,1\}}^n$$, \begin{align*} &\text{BernoulliLogitGLM}(y~|~x, \alpha, \beta) = \prod_{1\leq i \leq n}\text{Bernoulli}(y_i~|~\text{logit}^{-1}(\alpha_i + x_i\cdot \beta))\\ &= \prod_{1\leq i \leq n} \left\{ \begin{array}{ll} \text{logit}^{-1}(\alpha_i + \sum_{1\leq j\leq m}x_{ij}\cdot \beta_j) & \text{if } y_i = 1, \text{ and} \\ 1 - \text{logit}^{-1}(\alpha_i + \sum_{1\leq j\leq m}x_{ij}\cdot \beta_j) & \text{if } y_i = 0. \end{array} \right. \end{align*}

### Distribution statement

y ~ bernoulli_logit_glm(x, alpha, beta)

Increment target log probability density with bernoulli_logit_glm_lupmf(y | x, alpha, beta).

Available since 2.25

### Stan Functions

real bernoulli_logit_glm_lpmf(int y | matrix x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

Available since 2.23

real bernoulli_logit_glm_lupmf(int y | matrix x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.25

real bernoulli_logit_glm_lpmf(int y | matrix x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

Available since 2.23

real bernoulli_logit_glm_lupmf(int y | matrix x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.25

real bernoulli_logit_glm_lpmf(array[] int y | row_vector x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

Available since 2.23

real bernoulli_logit_glm_lupmf(array[] int y | row_vector x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.25

real bernoulli_logit_glm_lpmf(array[] int y | row_vector x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

Available since 2.23

real bernoulli_logit_glm_lupmf(array[] int y | row_vector x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.25

real bernoulli_logit_glm_lpmf(array[] int y | matrix x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

Available since 2.18

real bernoulli_logit_glm_lupmf(array[] int y | matrix x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.25

real bernoulli_logit_glm_lpmf(array[] int y | matrix x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

Available since 2.18

real bernoulli_logit_glm_lupmf(array[] int y | matrix x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

Available since 2.25

array[] int bernoulli_logit_glm_rng(matrix x, vector alpha, vector beta)
Generate an array of Bernoulli variates with chances of success inv_logit(alpha + x * beta); may only be used in transformed data and generated quantities blocks.

Available since 2.29

array[] int bernoulli_logit_glm_rng(row_vector x, vector alpha, vector beta)
Generate an array of Bernoulli variates with chances of success inv_logit(alpha + x * beta); may only be used in transformed data and generated quantities blocks.

Available since 2.29