BayesianRidge

#include <Skigen/LinearModel>

template <typename Scalar = double>
class Skigen::BayesianRidge(max_iter=300, tol=1e-3, alpha_1=1e-6, alpha_2=1e-6, lambda_1=1e-6, lambda_2=1e-6, alpha_init=std::nullopt, lambda_init=std::nullopt, compute_score=false, fit_intercept=true, copy_X=true, verbose=false)

Bayesian ridge regression.

Fit a Bayesian ridge model with Gamma hyper-priors on the noise precision $\alpha$ and the weight precision $\lambda$. The implementation follows Appendix A of (Tipping, 2001) with MacKay's (1992) update rules for the regularization parameters.

Mirrors sklearn.linear_model.BayesianRidge.

The objective maximised by the iterative procedure is the (log) marginal likelihood

$$\log p(y \mid \alpha, \lambda) = \tfrac{1}{2}\bigl[ N\log\alpha + p\log\lambda - \alpha \lVert y - Xw \rVert^2 - \lambda \lVert w \rVert^2 + \log |\Sigma| - N \log 2\pi \bigr] + \mathrm{const.}$$

---

---

Attributes:

• max_iter : int

• tol : Scalar

• alpha_1 : Scalar

• alpha_2 : Scalar

• lambda_1 : Scalar

• lambda_2 : Scalar

• alpha_init : std::optional< Scalar >

• lambda_init : std::optional< Scalar >

• compute_score : bool

• fit_intercept : bool

• verbose : bool

• coef : RowVectorType Posterior mean of the weights, shape (1 × n_features).

• intercept : Scalar Independent term in the decision function.

• alpha : Scalar Estimated noise precision $\alpha$.

• lambda_ : Scalar Estimated weight precision $\lambda$.

• sigma : MatrixType Posterior covariance of the weights, shape (n_features × n_features).

• scores : VectorType Log-marginal-likelihood values recorded during fitting.

• n_iter : int Number of iterations executed.

• X_offset : RowVectorType Centering offsets used for X (zeros if fit_intercept=false).

• X_scale : RowVectorType Per-feature scaling (always ones in this implementation).

• y_offset : Scalar Centering offset used for y.

---

Methods

SKIGEN_PARAMS()

Fit the Bayesian Ridge model via SVD-based evidence maximisation.

---

fit(X, y)

Fit from a sparse design matrix (densifies internally).

---

predict(X)

Predictive mean $\hat{y} = X w + b$.

---

score(X, y)

$R^2$ score.

---

predict(X)

Predictive mean and standard deviation.

Tag-dispatch overload selected by passing Skigen::with_std. Returns {y_mean, y_std}. The standard deviation is the marginal predictive std-dev: $\sigma_{\hat{y}}^2(x_*) = x_*^\top \Sigma\, x_* + 1/\alpha$.

---

predict(X)

Tag overload (with_std) — calls predict(X, std::true_type{}).

---

Example

Skigen::BayesianRidge<double> model(
    /*max_iter=*/300, /*tol=*/1e-3,
    /*alpha_1=*/1e-6, /*alpha_2=*/1e-6,
    /*lambda_1=*/1e-6, /*lambda_2=*/1e-6,
    /*alpha_init=*/std::nullopt, /*lambda_init=*/std::nullopt,
    /*compute_score=*/true);
model.fit(X, y);

std::cout << "=== BayesianRidge ===\n";
std::cout << "  alpha (noise precision)  = " << model.alpha() << "\n";
std::cout << "  lambda (weight precision)= " << model.lambda_() << "\n";
std::cout << "  n_iter = " << model.n_iter() << "\n";
std::cout << "  coef   = " << model.coef() << "\n";
std::cout << "  intercept = " << model.intercept() << "\n";
std::cout << "  R^2 (train) = " << model.score(X, y) << "\n";

// Predict with confidence intervals on the first 5 samples.
Eigen::MatrixXd X_test = X.topRows(5);
auto [y_mean, y_std] = model.predict(X_test, Skigen::with_std);
std::cout << "\nPredictions with 95% CI (first 5 rows):\n";
for (Eigen::Index i = 0; i < y_mean.size(); ++i) {
    std::cout << "  y_pred = " << std::setw(7) << y_mean(i)
              << "  +/- " << std::setw(6) << 1.96 * y_std(i)
              << "  (true=" << y(i) << ")\n";
}

---