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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
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