16.2 Normal-Id Generalized Linear Model (Linear Regression)
Stan also supplies a single function for a generalized linear lodel with normal likelihood and identity link function, i.e. a function for a linear regression. This provides a more efficient implementation of linear regression than a manually written regression in terms of a normal likelihood and matrix multiplication.
16.2.1 Probability Distribution Function
If \(x\in \mathbb{R}^{n\cdot m}, \alpha \in \mathbb{R}^n, \beta\in \mathbb{R}^m, \sigma\in \mathbb{R}^+\), then for \(y \in \mathbb{R}^n\), \[ \text{NormalIdGLM}(y|x, \alpha, \beta, \sigma) = \prod_{1\leq i \leq n}\text{Normal}(y_i|\alpha_i + x_i\cdot \beta, \sigma). \]
16.2.2 Sampling Statement
y ~ normal_id_glm(x, alpha, beta, sigma)
Increment target log probability density with normal_id_glm_lupdf(y | x, alpha, beta, sigma).
16.2.3 Stan Functions
real normal_id_glm_lpdf(real y | matrix x, real alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma.
real normal_id_glm_lupdf(real y | matrix x, real alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma dropping constant additive terms.
real normal_id_glm_lpdf(real y | matrix x, vector alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma.
real normal_id_glm_lupdf(real y | matrix x, vector alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma dropping constant additive terms.
real normal_id_glm_lpdf(vector y | row_vector x, real alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma.
real normal_id_glm_lupdf(vector y | row_vector x, real alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma dropping constant additive terms.
real normal_id_glm_lpdf(vector y | row_vector x, vector alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma.
real normal_id_glm_lupdf(vector y | row_vector x, vector alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma dropping constant additive terms.
real normal_id_glm_lpdf(vector y | matrix x, real alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma.
real normal_id_glm_lupdf(vector y | matrix x, real alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma dropping constant additive terms.
real normal_id_glm_lpdf(vector y | matrix x, vector alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma.
real normal_id_glm_lupdf(vector y | matrix x, vector alpha, vector beta, real sigma)
The log normal probability density of y given location alpha + x * beta
and scale sigma dropping constant additive terms.