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## 15.2 Normal-Id Generalised Linear Model (Linear Regression)

Stan also supplies a single primitive for a Generalised Linear Model with normal likelihood and identity link function, i.e. a primitive for a linear regression. This should provide a more efficient implementation of linear regression than a manually written regression in terms of a normal likelihood and matrix multiplication.

### 15.2.1 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 $$y \in \mathbb{R}^n$$, $\text{NormalIdGLM}(y|x, \alpha, \beta, \sigma) = \prod_{1\leq i \leq n}\text{Normal}(y_i|\alpha_i + x_i\cdot \beta, \sigma).$

### 15.2.2 Sampling Statement

y ~ normal_id_glm(x, alpha, beta, sigma)

Increment target log probability density with normal_id_glm_lpdf(y | x, alpha, beta, sigma) dropping constant additive terms.

### 15.2.3 Stan Functions

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, where a constant intercept alpha and sigma is used for all observations. The number of rows of the independent variable matrix x needs to match the length of the dependent variable vector y and the number of columns of x needs to match the length of the weight vector beta.

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, where a constant sigma is used for all observations and 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 length of the dependent variable vector y and the number of columns of x needs to match the length of the weight vector beta.