Binomial

Binomial defaults to the logit link and the averaged Bernoulli negative log likelihood:

F(y,\eta) = \frac1n\sum_{i=1}^n \left[ \log(1 + e^{\eta_i}) - y_i\eta_i \right], \qquad p_i = \frac{1}{1 + e^{-\eta_i}}.

The predictor derivatives are

\frac{\partial F}{\partial\eta_i} = \frac{p_i - y_i}{n}, \qquad \frac{\partial^2 F}{\partial\eta_i^2} = \frac{p_i(1 - p_i)}{n} \leq \frac{1}{4n}.

The implementation evaluates the loss and derivatives stably for large logits. Responses must be finite and lie in $[0,1]$ . Fractional values represent proportions with equal observation weights; this API does not accept trial counts or arbitrary observation weights. The intercept is fitted by default.

Evaluate the loss

use shrinkage::{Binomial, Datafit};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let y = [0.0, 1.0];
    let datafit = Binomial::new(&y);
    assert!((datafit.value(&[0.0, 0.0])? - 2.0_f64.ln()).abs() < 1e-14);
    Ok(())
}

Fitted predictions are logits. Apply the logistic function to obtain probabilities. See logistic regression for a stable conversion and a complete fit.

Proximal gradient supports the built-in proximal penalties. IRLS supports L1, ridge, and elastic net. The hybrid solver supports clustered lasso, chain fused lasso, and SLOPE. Unpenalized separated data may have no finite minimizer; inspect termination and choose a regularized model when appropriate.

Use .with_link(Probit) or a custom Rust link for IRLS fitting and paths. Predictions then use the selected predictor scale; call the datafit's .inverse_link(eta) to obtain probabilities.

API: shrinkage::Binomial.