# Gaussian

`Gaussian` fits a single real-valued response using averaged squared error: 

$$

  F(y,\eta) = \frac{1}{2n}\sum_{i=1}^n(y_i - \eta_i)^2, \qquad \nabla_\eta F
            = \frac{\eta - y}{n},\qquad \nabla_\eta^2 F = \frac{1}{n}I.

$$

Supply one finite response per observation. The constructor borrows the response slice. By default, the solver fits a free, unpenalized intercept; use `.fit_intercept(false)` to disable it. Predictions are fitted response means. 

<a id="evaluate-the-loss"></a>

## Evaluate the loss

```rust
use shrinkage::{Datafit, Gaussian};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let y = [1.0, 3.0];
    let datafit = Gaussian::new(&y);
    assert!((datafit.value(&[0.0, 2.0])? - 0.5).abs() < 1e-14);
    Ok(())
}
```

For model fitting, see [linear regression](<../models/linear>) or the [Lasso convenience API](<../models/lasso>). All four built-in solvers support Gaussian problems with their respective compatible penalties. Because the loss averages over observations, a penalty strength is not multiplied by the sample size. See the [shared conventions](<../modeling>). 

The default identity link makes predictors equal response means. Use `.with_link(...)` to select a different Rust link for IRLS fitting and paths; convert predictions with `.inverse_link(eta)`. 

API: [`shrinkage::Gaussian`](<https://shrinkage.cc/packages/rust/reference/8f2aa89a892fb708e0e09c276910730b91ed3cb252a10cacd0a458cfbe59be13>).