Unbounded Discrete Distributions
The unbounded discrete distributions have support over the natural numbers (i.e., the non-negative integers).
Negative binomial distribution
For the negative binomial distribution Stan uses the parameterization described in Gelman et al. (2013). For alternative parameterizations, see section negative binomial glm.
Probability mass function
If \(\alpha \in \mathbb{R}^+\) and \(\beta \in \mathbb{R}^+\), then for \(n \in \mathbb{N}\), \[\begin{equation*} \text{NegBinomial}(n~|~\alpha,\beta) = \binom{n + \alpha - 1}{\alpha - 1} \, \left( \frac{\beta}{\beta+1} \right)^{\!\alpha} \, \left( \frac{1}{\beta + 1} \right)^{\!n} \!. \end{equation*}\]
The mean and variance of a random variable \(n \sim \text{NegBinomial}(\alpha,\beta)\) are given by \[\begin{equation*} \mathbb{E}[n] = \frac{\alpha}{\beta} \ \ \text{ and } \ \ \text{Var}[n] = \frac{\alpha}{\beta^2} (\beta + 1). \end{equation*}\]
Distribution statement
n ~ neg_binomial(alpha, beta)
Increment target log probability density with neg_binomial_lupmf(n | alpha, beta).
Stan functions
real neg_binomial_lpmf(ints n | reals alpha, reals beta)
The log negative binomial probability mass of n given shape alpha and inverse scale beta
real neg_binomial_lupmf(ints n | reals alpha, reals beta)
The log negative binomial probability mass of n given shape alpha and inverse scale beta dropping constant additive terms
real neg_binomial_cdf(ints n | reals alpha, reals beta)
The negative binomial cumulative distribution function of n given shape alpha and inverse scale beta
real neg_binomial_lcdf(ints n | reals alpha, reals beta)
The log of the negative binomial cumulative distribution function of n given shape alpha and inverse scale beta
real neg_binomial_lccdf(ints n | reals alpha, reals beta)
The log of the negative binomial complementary cumulative distribution function of n given shape alpha and inverse scale beta
R neg_binomial_rng(reals alpha, reals beta)
Generate a negative binomial variate with shape alpha and inverse scale beta; may only be used in transformed data and generated quantities blocks. alpha \(/\) beta must be less than \(2 ^ {29}\). For a description of argument and return types, see section vectorized function signatures.
Negative binomial distribution (alternative parameterization)
Stan also provides an alternative parameterization of the negative binomial distribution directly using a mean (i.e., location) parameter and a parameter that controls overdispersion relative to the square of the mean. Section combinatorial functions, below, provides a second alternative parameterization directly in terms of the log mean.
Probability mass function
The first parameterization is for \(\mu \in \mathbb{R}^+\) and \(\phi \in \mathbb{R}^+\), which for \(n \in \mathbb{N}\) is defined as \[\begin{equation*} \text{NegBinomial2}(n \, | \, \mu, \phi) = \binom{n + \phi - 1}{n} \, \left( \frac{\mu}{\mu+\phi} \right)^{\!n} \, \left( \frac{\phi}{\mu+\phi} \right)^{\!\phi} \!. \end{equation*}\]
The mean and variance of a random variable \(n \sim \text{NegBinomial2}(n~|~\mu,\phi)\) are \[\begin{equation*} \mathbb{E}[n] = \mu \ \ \ \text{ and } \ \ \ \text{Var}[n] = \mu + \frac{\mu^2}{\phi}. \end{equation*}\] Recall that \(\text{Poisson}(\mu)\) has variance \(\mu\), so \(\mu^2 / \phi > 0\) is the additional variance of the negative binomial above that of the Poisson with mean \(\mu\). So the inverse of parameter \(\phi\) controls the overdispersion, scaled by the square of the mean, \(\mu^2\).
Distribution statement
n ~ neg_binomial_2(mu, phi)
Increment target log probability density with neg_binomial_2_lupmf(n | mu, phi).
Stan functions
real neg_binomial_2_lpmf(ints n | reals mu, reals phi)
The log negative binomial probability mass of n given location mu and precision phi.
real neg_binomial_2_lupmf(ints n | reals mu, reals phi)
The log negative binomial probability mass of n given location mu and precision phi dropping constant additive terms.
real neg_binomial_2_cdf(ints n | reals mu, reals phi)
The negative binomial cumulative distribution function of n given location mu and precision phi.
real neg_binomial_2_lcdf(ints n | reals mu, reals phi)
The log of the negative binomial cumulative distribution function of n given location mu and precision phi.
real neg_binomial_2_lccdf(ints n | reals mu, reals phi)
The log of the negative binomial complementary cumulative distribution function of n given location mu and precision phi.
R neg_binomial_2_rng(reals mu, reals phi)
Generate a negative binomial variate with location mu and precision phi; may only be used in transformed data and generated quantities blocks. mu must be less than \(2 ^ {29}\). For a description of argument and return types, see section vectorized function signatures.
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 \(n \in \mathbb{N}\), \[\begin{equation*} \text{NegBinomial2Log}(n \, | \, \eta, \phi) = \text{NegBinomial2}(n | \exp(\eta), \phi). \end{equation*}\] 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.
Distribution statement
n ~ neg_binomial_2_log(eta, phi)
Increment target log probability density with neg_binomial_2_log_lupmf(n | eta, phi).
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.
Negative-binomial-2-log generalized linear model (negative binomial regression)
Stan also supplies a single function for a generalized linear model with negative binomial distribution and log link function, i.e. a function for a negative binomial regression. This provides a more efficient implementation of negative binomial regression than a manually written regression in terms of a negative binomial distribution and matrix multiplication.
Probability mass function
If \(x\in \mathbb{R}^{n\cdot m}, \alpha \in \mathbb{R}^n, \beta\in \mathbb{R}^m, \phi\in \mathbb{R}^+\), then for \(y \in \mathbb{N}^n\), \[\begin{equation*} \text{NegBinomial2LogGLM}(y~|~x, \alpha, \beta, \phi) = \prod_{1\leq i \leq n}\text{NegBinomial2}(y_i~|~\exp(\alpha_i + x_i\cdot \beta), \phi). \end{equation*}\]
Distribution statement
y ~ neg_binomial_2_log_glm(x, alpha, beta, phi)
Increment target log probability density with neg_binomial_2_log_glm_lupmf(y | x, alpha, beta, phi).
Stan functions
real neg_binomial_2_log_glm_lpmf(int y | matrix x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
real neg_binomial_2_log_glm_lupmf(int y | matrix x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
real neg_binomial_2_log_glm_lpmf(int y | matrix x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
real neg_binomial_2_log_glm_lupmf(int y | matrix x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
real neg_binomial_2_log_glm_lpmf(array[] int y | row_vector x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
real neg_binomial_2_log_glm_lupmf(array[] int y | row_vector x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
real neg_binomial_2_log_glm_lpmf(array[] int y | row_vector x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
real neg_binomial_2_log_glm_lupmf(array[] int y | row_vector x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
real neg_binomial_2_log_glm_lpmf(array[] int y | matrix x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
real neg_binomial_2_log_glm_lupmf(array[] int y | matrix x, real alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
real neg_binomial_2_log_glm_lpmf(array[] int y | matrix x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi.
real neg_binomial_2_log_glm_lupmf(array[] int y | matrix x, vector alpha, vector beta, real phi)
The log negative binomial probability mass of y given log-location alpha + x * beta and inverse overdispersion parameter phi dropping constant additive terms.
Poisson distribution
Probability mass function
If \(\lambda \in \mathbb{R}^+\), then for \(n \in \mathbb{N}\), \[\begin{equation*} \text{Poisson}(n|\lambda) = \frac{1}{n!} \, \lambda^n \, \exp(-\lambda). \end{equation*}\]
Distribution statement
n ~ poisson(lambda)
Increment target log probability density with poisson_lupmf(n | lambda).
Stan functions
real poisson_lpmf(ints n | reals lambda)
The log Poisson probability mass of n given rate lambda
real poisson_lupmf(ints n | reals lambda)
The log Poisson probability mass of n given rate lambda dropping constant additive terms
real poisson_cdf(ints n | reals lambda)
The Poisson cumulative distribution function of n given rate lambda
real poisson_lcdf(ints n | reals lambda)
The log of the Poisson cumulative distribution function of n given rate lambda
real poisson_lccdf(ints n | reals lambda)
The log of the Poisson complementary cumulative distribution function of n given rate lambda
R poisson_rng(reals lambda)
Generate a Poisson variate with rate lambda; may only be used in transformed data and generated quantities blocks. lambda must be less than \(2^{30}\). For a description of argument and return types, see section vectorized function signatures.
Poisson distribution, log parameterization
Stan also provides a parameterization of the Poisson using the log rate \(\alpha = \log \lambda\) as a parameter. This is useful for log-linear Poisson regressions so that the predictor does not need to be exponentiated and passed into the standard Poisson probability function.
Probability mass function
If \(\alpha \in \mathbb{R}\), then for \(n \in \mathbb{N}\), \[\begin{equation*} \text{PoissonLog}(n|\alpha) = \frac{1}{n!} \, \exp \left(n\alpha - \exp(\alpha) \right). \end{equation*}\]
Distribution statement
n ~ poisson_log(alpha)
Increment target log probability density with poisson_log_lupmf(n | alpha).
Stan functions
real poisson_log_lpmf(ints n | reals alpha)
The log Poisson probability mass of n given log rate alpha
real poisson_log_lupmf(ints n | reals alpha)
The log Poisson probability mass of n given log rate alpha dropping constant additive terms
R poisson_log_rng(reals alpha)
Generate a Poisson variate with log rate alpha; may only be used in transformed data and generated quantities blocks. alpha must be less than \(30 \log 2\). For a description of argument and return types, see section vectorized function signatures.
Poisson-log generalized linear model (Poisson regression)
Stan also supplies a single function for a generalized linear model with Poisson distribution and log link function, i.e. a function for a Poisson regression. This provides a more efficient implementation of Poisson regression than a manually written regression in terms of a Poisson 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 \(y \in \mathbb{N}^n\), \[\begin{equation*} \text{PoissonLogGLM}(y|x, \alpha, \beta) = \prod_{1\leq i \leq n}\text{Poisson}(y_i|\exp(\alpha_i + x_i\cdot \beta)). \end{equation*}\]
Distribution statement
y ~ poisson_log_glm(x, alpha, beta)
Increment target log probability density with poisson_log_glm_lupmf(y | x, alpha, beta).
Stan functions
real poisson_log_glm_lpmf(int y | matrix x, real alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta.
real poisson_log_glm_lupmf(int y | matrix x, real alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta dropping constant additive terms.
real poisson_log_glm_lpmf(int y | matrix x, vector alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta.
real poisson_log_glm_lupmf(int y | matrix x, vector alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta dropping constant additive terms.
real poisson_log_glm_lpmf(array[] int y | row_vector x, real alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta.
real poisson_log_glm_lupmf(array[] int y | row_vector x, real alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta dropping constant additive terms.
real poisson_log_glm_lpmf(array[] int y | row_vector x, vector alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta.
real poisson_log_glm_lupmf(array[] int y | row_vector x, vector alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta dropping constant additive terms.
real poisson_log_glm_lpmf(array[] int y | matrix x, real alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta.
real poisson_log_glm_lupmf(array[] int y | matrix x, real alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta dropping constant additive terms.
real poisson_log_glm_lpmf(array[] int y | matrix x, vector alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta.
real poisson_log_glm_lupmf(array[] int y | matrix x, vector alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta dropping constant additive terms.
Beta negative binomial distribution
Probability mass function
If \(r \in \mathbb{R}^+\), \(\alpha \in \mathbb{R}^+\), and \(\beta \in \mathbb{R}^+\), then for \(n \in \mathbb{N}\), \[\begin{equation*} \text{BetaNegBinomial}(n|r,\alpha,\beta) = \frac {\Gamma (n+r )}{n!\;\Gamma (r )} \frac {\mathrm {B} (\beta+n,\alpha +r )}{\mathrm {B} (\beta,\alpha )}. \end{equation*}\]
Distribution statement
n ~ beta_neg_binomial(r,alpha,beta)
Increment target log probability density with beta_neg_binomial_lupmf(n | r, alpha, beta).
Stan functions
real beta_neg_binomial_lpmf(ints n | reals r, reals alpha, reals beta)
The log beta negative binomial probability mass of n given parameters r, alpha and beta.
real beta_neg_binomial_lupmf(ints n | reals r, reals alpha, reals beta)
The log beta negative binomial probability mass of n given parameters r, alpha and beta dropping constant additive terms.
real beta_neg_binomial_cdf(ints n | reals r, reals alpha, reals beta)
The beta negative binomial cumulative distribution function of n given parameters r, alpha and beta.
real beta_neg_binomial_lcdf(ints n | reals r, reals alpha, reals beta)
The log of the beta negative binomial cumulative distribution function of n given parameters r, alpha and beta.
real beta_neg_binomial_lccdf(ints n | reals r, reals alpha, reals beta)
The log of the beta negative binomial complementary cumulative distribution function of n given parameters r, alpha and beta.
R beta_neg_binomial_rng(reals r, reals alpha, reals beta)
Generate a beta negative binomial variate with parameters r, alpha and beta; may only be used in transformed data and generated quantities blocks. r \(\cdot\) beta \(/\) (alpha\(-1\)) must be less than \(2 ^ {29}\). For a description of argument and return types, see section vectorized function signatures.