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 $y\in \{0,1\}$, $$\text{Bernoulli}(y|\theta )=\{\begin{array}{ll}\theta & \text{if }y=1,\text{ and}\\ 1-\theta & \text{if }y=0.\end{array}$$

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 $y\in \{0,1\}$, $$\text{BernoulliLogit}(y|\alpha )=\text{Bernoulli}(y|{\text{logit}}^{-1}(\alpha ))=\{\begin{array}{ll}{\text{logit}}^{-1}(\alpha )& \text{if }y=1,\text{ and}\\ 1-{\text{logit}}^{-1}(\alpha )& \text{if }y=0.\end{array}$$

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 $y\in {\{0,1\}}^n$, $$\begin{array}{rl}& \text{BernoulliLogitGLM}(y|x,\alpha ,\beta )=\prod _{1\le i\le n}\text{Bernoulli}(y_i|{\text{logit}}^{-1}({\alpha }_i+x_i\cdot \beta ))\\ & =\prod _{1\le i\le n}\{\begin{array}{ll}{\text{logit}}^{-1}({\alpha }_i+\sum _{1\le j\le m}{x}_{ij}\cdot {\beta }_j)& \text{if }{y}_i=1,\text{ and}\\ 1-{\text{logit}}^{-1}({\alpha }_i+\sum _{1\le j\le m}{x}_{ij}\cdot {\beta }_j)& \text{if }{y}_i=0.\end{array}\end{array}$$

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