code: from __future__ import print_function, division
  import numpy as np
  from scipy.stats import chi2, multivariate_normal
  from mlfromscratch.utils import mean_squared_error, train_test_split, polynomial_features
  
  
  
  class BayesianRegression(object):
      """Bayesian regression model. If poly_degree is specified the features will
      be transformed to with a polynomial basis function, which allows for polynomial
      regression. Assumes Normal prior and likelihood for the weights and scaled inverse
      chi-squared prior and likelihood for the variance of the weights.
  
      Parameters:
      -----------
      n_draws: float
          The number of simulated draws from the posterior of the parameters.
      mu0: array
          The mean values of the prior Normal distribution of the parameters.
      omega0: array
          The precision matrix of the prior Normal distribution of the parameters.
      nu0: float
          The degrees of freedom of the prior scaled inverse chi squared distribution.
      sigma_sq0: float
          The scale parameter of the prior scaled inverse chi squared distribution.
      poly_degree: int
          The polynomial degree that the features should be transformed to. Allows
          for polynomial regression.
      cred_int: float
          The credible interval (ETI in this impl.). 95 => 95% credible interval of the posterior
          of the parameters.
  
      Reference:
          https://github.com/mattiasvillani/BayesLearnCourse/raw/master/Slides/BayesLearnL5.pdf
      """
      def __init__(self, n_draws, mu0, omega0, nu0, sigma_sq0, poly_degree=0, cred_int=95):
          self.w = None
          self.n_draws = n_draws
          self.poly_degree = poly_degree
          self.cred_int = cred_int
  
          # Prior parameters
          self.mu0 = mu0
          self.omega0 = omega0
          self.nu0 = nu0
          self.sigma_sq0 = sigma_sq0
  
      # Allows for simulation from the scaled inverse chi squared
      # distribution. Assumes the variance is distributed according to
      # this distribution.
      # Reference:
      #   https://en.wikipedia.org/wiki/Scaled_inverse_chi-squared_distribution
      def _draw_scaled_inv_chi_sq(self, n, df, scale):
          X = chi2.rvs(size=n, df=df)
          sigma_sq = df * scale / X
          return sigma_sq
  
      def fit(self, X, y):
  
          # If polynomial transformation
          if self.poly_degree:
              X = polynomial_features(X, degree=self.poly_degree)
  
          n_samples, n_features = np.shape(X)
  
          X_X = X.T.dot(X)
  
          # Least squares approximate of beta
          beta_hat = np.linalg.pinv(X_X).dot(X.T).dot(y)
  
          # The posterior parameters can be determined analytically since we assume
          # conjugate priors for the likelihoods.
  
          # Normal prior / likelihood => Normal posterior
          mu_n = np.linalg.pinv(X_X + self.omega0).dot(X_X.dot(beta_hat)+self.omega0.dot(self.mu0))
          omega_n = X_X + self.omega0
          # Scaled inverse chi-squared prior / likelihood => Scaled inverse chi-squared posterior
          nu_n = self.nu0 + n_samples
          sigma_sq_n = (1.0/nu_n)*(self.nu0*self.sigma_sq0 + \
              (y.T.dot(y) + self.mu0.T.dot(self.omega0).dot(self.mu0) - mu_n.T.dot(omega_n.dot(mu_n))))
  
          # Simulate parameter values for n_draws
          beta_draws = np.empty((self.n_draws, n_features))
          for i in range(self.n_draws):
              sigma_sq = self._draw_scaled_inv_chi_sq(n=1, df=nu_n, scale=sigma_sq_n)
              beta = multivariate_normal.rvs(size=1, mean=mu_n[:,0], cov=sigma_sq*np.linalg.pinv(omega_n))
              # Save parameter draws
              beta_draws[i, :] = beta
  
          # Select the mean of the simulated variables as the ones used to make predictions
          self.w = np.mean(beta_draws, axis=0)
  
          # Lower and upper boundary of the credible interval
          l_eti = 50 - self.cred_int/2
          u_eti = 50 + self.cred_int/2
          self.eti = np.array([[np.percentile(beta_draws[:,i], q=l_eti), np.percentile(beta_draws[:,i], q=u_eti)] \
                                  for i in range(n_features)])
  
      def predict(self, X, eti=False):
  
          # If polynomial transformation
          if self.poly_degree:
              X = polynomial_features(X, degree=self.poly_degree)
  
          y_pred = X.dot(self.w)
          # If the lower and upper boundaries for the 95%
          # equal tail interval should be returned
          if eti:
              lower_w = self.eti[:, 0]
              upper_w = self.eti[:, 1]
              y_lower_pred = X.dot(lower_w)
              y_upper_pred = X.dot(upper_w)
              return y_pred, y_lower_pred, y_upper_pred
  
          return y_pred
  
apis: numpy.empty numpy.shape numpy.linalg.pinv scipy.stats.chi2.rvs numpy.percentile numpy.mean