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## 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.