def surrogate_model(n, dim, fun, fun_base, theta, X, y, F):
R, inv_R = correl.compute_R(theta, X)
beta_hat = np.linalg.inv(F.T @ inv_R @ F) @ F.T @ inv_R @ y
sigma2_hat = (y - F @ beta_hat).T @ inv_R @ (y - F @ beta_hat) / n
return R, inv_R, beta_hat, sigma2_hat
def logML(theta):
F = fun_base(X)
R, inv_R = correl.compute_R(theta, X)
beta_hat = np.linalg.inv(F.T @ inv_R @ F) @ F.T @ inv_R @ y
sigma2_hat = (y - F @ beta_hat).T @ inv_R @ (y - F @ beta_hat) / n
criterion = -0.5 * (n * np.log(sigma2_hat) + np.log(np.linalg.det(R)))
return -criterion
def distribFixe(theta):
""" compute the density of theta|sigma,X,beta"""
F = fun_base(X)
R, inv_R = correl.compute_R(theta, X)
More = (y - F @ beta).T @ inv_R @ (y - F @ beta) / (2 * sigma2_hat
) #beta is fixed here
P = np.exp(-(np.linalg.norm(theta - bias, 2)**2) /
(2 * V) - More) / np.sqrt(np.linalg.det(R))
return P
def distrib(theta):
""" compute the density of theta|sigma,X, here we compute beta as a deterministic function of the other parameter"""
F = fun_base(X)
R, inv_R = correl.compute_R(theta, X)
beta_hat = np.linalg.inv(
F.T @ inv_R @ F
) @ F.T @ inv_R @ y # beta is computed, he is not supposed to be fixed here
More = (y - F @ beta_hat).T @ inv_R @ (y - F @ beta_hat) / (2 * sigma2_hat)
P = np.exp(-(np.linalg.norm(theta - bias, 2)**2) /
(2 * V) - More) / np.sqrt(np.linalg.det(R))
return P
def logbayes(theta):
"""(alpha,delta) parameter for the invGammma prior on sigma,
(b,invGamma) parameter for the Gaussian prior on beta,
(th,V) for the standard gaussian prior on (th,V>0)"""
F = fun_base(X)
R, inv_R = correl.compute_R(theta, X)
beta_hat = np.linalg.inv(F.T @ inv_R @ F) @ F.T @ inv_R @ y
sigma2_hat = ((y - F @ beta_hat).T @ inv_R @ (y - F @ beta_hat) +
2 * delta) / (n + 2 * (alpha - 1)) #INVGAMA
beta_hat = np.linalg.inv(F.T @ inv_R @ F / sigma2_hat**2 + invGamma) @ (
invGamma @ b + F.T @ inv_R @ y / sigma2_hat**2)
More = 0.5 * (y - F @ beta_hat).T @ inv_R @ (
y - F @ beta_hat
) / sigma2_hat # we have to add this term because the expression of sigma2_hat does not imply further simplification as previouly whit the frequentist estimation of sigma2_hat
criterion = -0.5 * (n * np.log(sigma2_hat) + np.log(np.linalg.det(R)) +
np.linalg.norm(theta - th, 2)**2 / V) - More
return -criterion