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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 xRnm,αRn,βRm, then for yNn, PoisonLogGLM(y|x,α,β)=1inPoisson(yi|exp(αi+xiβ)).

14.7.2 Sampling statement

y ~ poisson_log_glm(x, alpha, beta)

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

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.

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(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(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(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(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(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.