14.7 Poisson-Log Generalized Linear Model (Poisson Regression)
Stan also supplies a single function for a generalized linear model with Poisson likelihood 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 likelihood and matrix multiplication.
14.7.1 Probability Mass Function
If \(x\in \mathbb{R}^{n\cdot m}, \alpha \in \mathbb{R}^n, \beta\in \mathbb{R}^m\), then for \(y \in \mathbb{N}^n\), \[ \text{PoisonLogGLM}(y|x, \alpha, \beta) = \prod_{1\leq i \leq n}\text{Poisson}(y_i|\exp(\alpha_i + x_i\cdot \beta)). \]
14.7.2 Sampling Statement
y ~ poisson_log_glm(x, alpha, beta)
Increment target log probability density with poisson_log_glm_lpmf(y | x, alpha, beta)
dropping constant additive terms.
14.7.3 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,
where the same intercept alpha and dependent variable value y are used for all observations.
The number of columns of x needs to match the size of the coefficient vector beta.
If x and y are data (not parameters) this function can be executed on a GPU.
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,
where an intercept alpha is used that is allowed to vary with the
different observations. Dependent variable value y is used for all observations.
The number of rows of x must match the size of alpha and
the number of columns of x needs to match the size of the coefficient vector beta.
If x and y are data (not parameters) this function can be executed on a GPU.
real poisson_log_glm_lpmf(int[] y | row_vector x, real alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta,
where the same intercept alpha is used for all observations.
The number of columns of x needs to match the size of the coefficient vector beta.
real poisson_log_glm_lpmf(int[] y | row_vector x, vector alpha, vector beta)
The log Poisson probability mass of y given the log-rate alpha + x * beta,
where an intercept alpha is used that is allowed to vary with the
different observations.
The number of rows of x must match the size of alpha and
the number of columns of x needs to match the size of the coefficient vector beta.
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,
where the same intercept alpha is used for all observations. The
number of rows of the independent variable matrix x needs to match
the size of the dependent variable vector y and the number of
columns of x needs to match the size of the coefficient vector beta.
If x and y are data (not parameters) this function can be executed on a GPU.
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,
where an intercept alpha is used that is allowed to vary with the
different observations. The number of rows of the independent variable
matrix x needs to match the size of the dependent variable vector
y and alpha and the number of columns of x needs to match the size of the
coefficient vector beta.
If x and y are data (not parameters) this function can be executed on a GPU.